N = 2 - uni-jena.de · 11.15.Ha, 11.10.Gh Keyw ords: sup ersymmetry, lattice mo dels 1 Supp orted b...

Low-dimensional Supersymmetri13 Latti13e Models1G Bergner T Kaestner S Uhlmann and A Wipf2Theoretis13h-Physikalis13hes InstitutFriedri13h-S13hiller-Universitaumlt JenaFroumlbelstieg 1 D-07743 Jena GermanyAbstra13tWe study and simulate N = 2 supersymmetri13 Wess-Zumino models in one and two di-mensions For any 13hoi13e of the latti13e derivative the theories 13an be made manifestlysupersymmetri13 by adding appropriate improvement terms 13orresponding to dis13retiza-tions of surfa13e integrals In one dimension our simulations show that a model withthe Wilson derivative and the Stratonovit13h pres13ription for this dis13retization leadsto far better results at nite latti13e spa13ing than other models with Wilson fermions13onsidered in the literature In parti13ular we 13he13k that fermioni13 and bosoni13 masses13oin13ide and the unbroken Ward identities are fullled to high a1313ura13y Equally goodresults for the ee13tive masses 13an be obtained in a model with the SLAC derivative(even without improvement terms)In two dimensions we introdu13e a non-standard Wilson term in su13h a way that thedis13retization errors of the kineti13 terms are only of order O(a4) Masses extra13tedfrom the 13orresponding manifestly supersymmetri13 model prove to approa13h their 13on-tinuum values mu13h qui13ker than those from a model 13ontaining the standard Wilsonterm Again a 13omparable enhan13ement 13an be a13hieved in a theory using the SLACderivativePACS numbers 1130Pb 1260Jv 1115Ha 1110GhKeywords supersymmetry latti13e models1Supported by the Deuts13he Fors13hungsgemeins13haft DFG-Wi 7778-22email GBergner TKaestner AWipftpiuni-jenade and SUhlmannuni-jenade

1 Introdu13tionSupersymmetry is an important ingredient of modern high energy physi13s beyond the stan-dard model sin13e boson masses are prote13ted by supersymmetry in su13h theories with 13hiralfermions it helps to redu13e the hierar13hy and ne-tuning problems drasti13ally and withingrand unied theories it leads to predi13tions of the proton life-time in agreement withpresent day experimental bounds As low energy physi13s is manifestly not supersymmet-ri13 this symmetry has to be broken at some energy s13ale However non-renormalizationtheorems in four dimensions ensure that tree level supersymmetri13 theories preserve super-symmetry at any nite order of perturbation theory therefore supersymmetry has to bebroken non-perturbatively [1The latti13e formulation of quantum eld theories provides a systemati13 tool to investigatenon-perturbative problems In the 13ase of supersymmetri13 eld theories their formulation ishampered by the fa13t that the supersymmetry algebra 13loses on the generator of innitesimaltranslations [2 Sin13e Poin13areacute symmetry is expli13itly broken by the dis13retization oneis tempted to modify the supersymmetry algebra so as to 13lose on dis13rete translationsHowever as latti13e derivatives do not satisfy the Leibniz rule supersymmetri13 a13tions forintera13ting theories will in general not be invariant under su13h latti13e supersymmetries Theviolation of the Leibniz rule is an O(a) ee13t and supersymmetry naively will therefore berestored in the 13ontinuum limit In the 13ase of Poin13areacute symmetry the dis13rete remnants ofthe symmetry on the latti13e are su13ient to prohibit the appearan13e of relevant operators inthe ee13tive a13tion whi13h are invariant only under a subset of the Poin13areacute group and requirene tuning of their 13oe13ients in order to arrive at an invariant 13ontinuum limit In generi13latti13e formulations there are no dis13rete remnants of supersymmetry transformations onthe latti13e in su13h theories supersymmetry in the 13ontinuum limit 13an only be a13hieved byappropriately ne-tuning the bare 13ouplings of all supersymmetry-breaking 13ounterterms [3An additional 13ompli13ation in the formulation of supersymmetri13 theories on the latti13e isthe fermion doubling problem Lo13al and translationally invariant hermitean Dira13 opera-tors on the latti13e automati13ally des13ribe fermions of both 13hiralities [4 5 the fermioni13extra degrees of freedom are usually not paired with bosoni13 modes and so lead to supersym-metry breaking Generi13 pres13riptions eliminating these extra fermioni13 modes also breaksupersymmetryAs a simple supersymmetri13 theory the Wess-Zumino model in two dimensions has beenthe subje13t of intensive analyti13 and numeri13al investigations Early attempts in13luded1

the 13hoi13e of the nonlo13al SLAC derivative (thereby avoiding the doubling problem) in theN = 1 version of this model [6 7 8 This was motivated by the idea that the Fouriertransform of the latti13e theory should 13oin13ide with that of the 13ontinuum theory A Hamil-tonian approa13h where the SLAC derivative minimizes supersymmetry-breaking artifa13tsintrodu13ed by non-antisymmetri13 latti13e derivatives was dis13ussed in [9 Alternatively alo13al a13tion with Wilson fermions was 13onstru13ted at the 13ost of a nonlo13al supersymme-try variation [103 simulations of this model [17 indi13ate that this theory with a 13ubi13superpotential indeed features non-perturbative supersymmetry breakingIn order to manifestly preserve some subalgebra of the N = 1 supersymmetry algebra on thelatti13e the above dis13ussion suggests to 13hoose a subalgebra independent of the momentumoperator Thus super13harges Q+ and Qminus 13an be dened using (non)lo13al derivatives on aspatial latti13e The 13hoi13e of a 13ontinuous time then allows for a Hamiltonian dened byH = Q2

+ = Q2minus whi13h 13ommutes with the super13harges [18 19 and automati13ally 13ontainsa Wilson term This strategy 13an be generalized to the N = 2 model on a spatial latti13ean analysis shows that a subalgebra admitting an O(2) R-symmetry 13an be preserved [20Simulations for the N = 1 (see [21) and the N = 2 model [22 have been done using thelo13al Hamiltonian Monte-Carlo methods Further simulations based on the Green fun13tionMonte-Carlo method for N = 1 indi13ate that supersymmetry is unbroken for a quarti13superpotential and has a broken and an unbroken phase for 13ubi13 superpotentials [23 24With a dis13rete time-13oordinate one has to resort to a13tion- rather than Hamiltonian-basedapproa13hes The perfe13t a13tion approa13h was pursued for the free N = 1 theory in [25a generalization for intera13ting theories seems however problemati13 A treatment of the

N = 2 Wess-Zumino model in the Dira13-Kaumlhler formalism preserves a s13alar supersymmetryon the latti13e but leads to non-13onjugate transformations of the 13omplex s13alar eld andits 13onjugate This enlarges the spa13e of states and presumably renders the theory non-unitary [10 A related [26 idea avoiding these problems goes ba13k to the idea of Ni13olai [27that (in this 13ase) s13alar supersymmetri13 eld theories admit new bosoni13 variables with aJa13obian 13an13elling the determinant from integrating out the fermions in terms of whi13h the3A similar approa13h with staggered fermions leads to problems in the 13ontinuum limit [11 In thefour-dimensional model with N = 1 supersymmetry with Ginsparg-Wilson fermions latti13e 13hiral symmetryis in13ompatible with Yukawa 13ouplings [12 13 however the theory 13an be regularized by supersymmetri13higher derivative 13orre13tions whi13h leads to a supersymmetri13 13ontinuum limit within perturbation the-ory [14 It 13ould be shown that supersymmetri13 Ward identities are satised up to order g2 in the 13oupling13onstant [15 16 2

bosoni13 a13tion is purely Gaussian In general the new variables are 13ompli13ated nonlo13alfun13tions of the original ones for N = 2 supersymmetry however lo13al Ni13olai variables13an be expe13ted [28 Starting from a dis13retized form of the Ni13olai variables found in the13ontinuum one simply denes the bosoni13 part of the a13tion as the sum over squares of theNi13olai variables the fermioni13 part of the a13tion is adjusted so that the determinant of thefermion matrix still 13an13els the Ja13obian This leads to an a13tion manifestly preserving partof the 13ontinuum supersymmetry on the latti13e and whi13h 13ontains improvement terms inaddition to the naive latti13ization of the 13ontinuum a13tion [28 20 As stated in [29 30 thebreaking of some of the 13ontinuum supersymmetries 13an in this framework be tra13ed ba13k tothe fa13t that the improvement terms are not 13ompatible with ree13tion positivity Howeverthe violation of Osterwalder-S13hrader positivity is an O(a2gphys) ee13t and thus should benegligible at least at weak 13oupling [31 An analysis of the perturbation series shows thatthese terms in an o-shell formulation of the theory with a 13ubi13 superpotential and Wilsonfermions lead to tadpole diagrams whi13h diverge linearly in the 13ontinuum limit [32 A13an13ellation between these would-be 13entral 13harge terms and the naive dis13retization ofthe 13ontinuum Hamiltonian has been suggested as a solution for the 13onundrum raisedin [29 that the latti13e result of the number of zero-modes of the Dira13 operator seems todier vastly from the 13ontinuum answer [9The possibility to introdu13e Ni13olai variables is 13losely related to the fa13t that the (2 2)Wess-Zumino model is a topologi13al theory of Witten type i e the a13tion is of the formS = QΛ for a s13alar super13harge Q on13e auxiliary elds are introdu13ed [33 34 Thisformulation manifestly preserves Q-supersymmetry on the latti13e and therefore guaranteesthat the theory remains supersymmetri13 in the limit of vanishing latti13e spa13ing Latti13etheories of this type in various dimensions and with dierent degrees of supersymmetryhave been 13lassied in [35 It turns out that in this formulation some remnant of the U(1)VR-symmetry whi13h is left unbroken by the superpotential at the 13lassi13al level but brokenby the Wilson terms is restored in the 13ontinuum limit even non-perturbatively [31 thisalso indi13ates that at least for the 13ubi13 superpotential under 13onsideration supersymmetryis not broken nonperturbativelyAt a xed point in spa13e the Wess-Zumino model redu13es to supersymmetri13 quantumme13hani13s A naive dis13retization of the a13tion with a Wilson term for just the fermionleads to diering masses for fermions and bosons in the 13ontinuum limit [36 This 13an betra13ed ba13k [37 to 1-loop 13ontributions (of UV degree 0) of the fermion doublers to the bosonpropagator whi13h have to be 13an13elled by an appropriate 13ounterterm that was negle13ted3

in [36 As soon as the a13tion is amended by a Wilson term also for the boson [36 anadditional diagram involving the boson doublers 13an13els the nite 13orre13tion of the fermiondoublers and no 13ounterterms are required to a13hieve the desired 13ontinuum limit Inthis 13ase fermion and boson masses agree in the limit of vanishing latti13e spa13ing andsupersymmetri13 Ward identities are fullled to great a1313ura13yAs expe13ted for a theory with N = 2 supersymmetry supersymmetri13 quantum me13hani13s13an be formulated in terms of lo13al Ni13olai variables [28 However this a13tion diers fromthe amended form of [36 by a further intera13tion term whi13h be13omes an integral over atotal derivative in the 13ontinuum limitIn this paper we s13rutinize six dierent latti13e a13tions for supersymmetri13 quantum me13han-i13s based on Wilson and SLAC fermions with and without su13h improvement terms Thethree models without improvement terms are not manifestly supersymmetri13 in the pres-en13e of intera13tions and dier by bosoni13 terms whi13h be13ome irrelevant in the 13ontinuumlimit as well as by the 13hoi13e of the latti13e derivative the other three models with manifestsupersymmetry dier in the pres13ription for the evaluation of the improvement term andalso in the 13hoi13e of the latti13e derivative they 13an be 13onstru13ted from three dierentNi13olai variables We 13ompare the ee13tive masses for intera13ting theories with a quarti13superpotential and analyze the Ward identities of the broken and unbroken supersymmetriesat various latti13e sizes and 13ouplings The 13entral observation here is that the manifestlysupersymmetri13 theory with the so-13alled Stratonovi13h pres13ription [38 for the evaluationof the improvement term (whi13h is the dis13retization of a 13ontinuum surfa13e integral) leadsto far better results than the model with an Ito pres13ription at nite latti13e spa13ing Themass extra13tion for the models with SLAC derivative is at rst sight hampered by an os-13illating behavior for nearby insertions in the bosoni13 and fermioni13 two-point fun13tionshowever this Gibbs phenomenon is under good analyti13al 13ontrol and 13an be softened bythe appli13ation of an appropriate lter With the help of this optimal lter the resultseven surpass those of the model with Stratonovi13h pres13riptionAlong the way we show for whi13h superpotentials and derivatives one 13an guarantee posi-tivity of the fermion determinants In those 13ases where the determinant 13an be 13omputedexa13tly we analyze under whi13h 13ir13umstan13es they 13onverge to the 13orre13t 13ontinuum re-sults these exa13t results are 13ru13ial for our simulations Again the model with Stratonovi13hpres13ription is ahead of the one with Ito pres13ription the former leads to a determinant withthe 13orre13t 13ontinuum limit whereas the latter diers from it by a fa13tor whi13h depends onthe superpotential 4

Furthermore we study three dierent manifestly supersymmetri13 dis13retizations of the N=2Wess-Zumino model in two dimensions Instead of trying to generalize the Stratonovi13h pre-s13ription to two dimensions we introdu13e a non-standard Wilson term (13orresponding toan imaginary Wilson parameter in the holomorphi13 superpotential) in su13h a way that thedis13retization errors for the eigenvalues of the free (bosoni13 and fermioni13) kineti13 operatorsare only of order O(a4) instead of order O(a) for the standard Wilson term In the simula-tions we study the ee13t of the resulting violation of ree13tion positivity and 13ompare theresults with those of the model with SLAC fermions Due to 13al13ulational 13onstraints wehave to restri13t the 13omputations to smaller and intermediate values of the 13ouplingAs a theoreti13al ba13kground we show that the dis13retized Wess-Zumino model in two di-mensions with the SLAC derivative has a renormalizable 13ontinuum limitThe paper is organized as follows In se13tion 2 we introdu13e the quantum me13hani13al mod-els on the latti13e with and without improvement terms and dis13uss whi13h behavior of theintera13ting theories at nite latti13e spa13ings 13an be gathered from their (non-)invarian13eunder supersymmetry transformations in the free 13ase In subse13tion 24 we give detailsabout the positivity of the fermion determinants and derive their respe13tive 13ontinuum lim-its The results of the ee13tive masses and the Ward identities 13an be found in se13tion 3 Inse13tion 4 we dis13uss the dis13retizations of the N = 2 Wess-Zumino model in two dimensionsand present the results of the mass extra13tions Se13tion 5 13overs the algorithmi13 aspe13ts ofour simulations in13luding a derivation of the Gibbs phenomenon for the SLAC 13orrelatorswhi13h justies the appli13ation of the lter for the mass extra13tion in the quantum me13hani-13al model Finally se13tion 6 13ontains the proof that the Wess-Zumino model on the latti13ewith the SLAC derivative is renormalizable to rst order in perturbation theory a te13hni13alpart of this proof is 13ompleted in appendix A2 Supersymmetri13 quantum me13hani13sIn this se13tion we introdu13e the a13tion for supersymmetri13 quantum me13hani13s in a lan-guage whi13h 13an be easily generalized later on to the two-dimensional Eu13lidean Wess-Zumino model In subse13tions 221 and 231 we present six dierent latti13e versions of the13ontinuum theory with Eu13lidean a13tionScont =

int

dτ(1

2φ2 +

1

2W prime2 + ψψ + ψW primeprimeψ

)

where W prime(φ) equiv dW (φ)

dφ (1)5

The 13ontinuum model is invariant under two supersymmetry transformationsδ(1)φ = εψ δ(1)ψ = minusε(φ+W prime) δ(1)ψ = 0

δ(2)φ = ψε δ(2)ψ = 0 δ(2)ψ = (φminusW prime)ε(2)with anti13ommuting parameters ε and ε The latti13e approximations 13onsidered below dierby the 13hoi13e of the latti13e derivative andor the dis13retization pres13ription for a 13ontinuumsurfa13e term in subse13tion 24 we argue why three of them lead to far better approximationsto the 13ontinuum theory The results of our simulations for a quarti13 superpotentialW (φ) =

m2φ2 + g

4φ4 (with positive m g) are dis13ussed in se13tion 321 Latti13e modelsWe start from a one-dimensional periodi13 time latti13e Λ with real bosoni13 variables φx andtwo sets of real Grayumlmann variables ψx ψx on the latti13e sites x isin Λ = 1 N Theintegral and 13ontinuum derivative in (1) are repla13ed by a Riemann sum a

sum and a latti13ederivative part where a denotes the latti13e 13onstant Two dierent antisymmetri13 latti13ederivatives will be used in what follows These are the ultralo13al derivativepartxy =

1

2a(δx+1y minus δxminus1y) (3)with doublers4 and the nonlo13al SLAC derivative without doublers whi13h for an odd number

N of latti13e sites takes the form [39 9part slacx 6=y =

(minus1)xminusy

a

πN

sin(π(xminus y)N)and part slac

xx = 0 (4)If we allow for non-antisymmetri13 derivatives then we may add a multiple of the symmetri13latti13e Lapla13ian∆xy =

1

a2(δx+1y minus 2δxy + δxminus1y) (5)to part to get rid of the doublers In this way we obtain a one-parameter family of ultralo13alderivatives with Wilson term

part minus 12ar∆ minus1 le r le 1 (6)4It should be noted that the symmetri13 13ombination of forward and ba13kward derivatives leading to (3)yields an antisymmetri13 matrix (partxy) 6

interpolating between the forward (or right) derivative for r = minus1 and the ba13kward (orleft) derivative for r = 1 As will be argued in subse13tion 24 for the quarti13 superpotentialW (φ) = m

2φ2 + g

4φ4 with positive parameters m and g to be 13onsidered below we shall needthe latter with matrix elements

partbxy = partxy minus

a

2∆xy =

1

a(δxy minus δxminus1y) (7)The ba13kward derivative is free of doublers and not antisymmetri13 For periodi13 boundary13onditions the derivative operators part slac part and partb are all given by 13ir13ulant matri13es whi13h13ommute with ea13h other22 Latti13e models without improvementA straightforward dis13retization of the 13ontinuum a13tion (1) would be

Snaive =a

2

sum

x

(

(partφ)2x +W 2

x

)

+ asum

xy

ψx (partxy +Wxy)ψy (8)where we s13rutinize below three possibilities for the latti13e derivatives part as well as forthe terms Wx Wxy derived from the superpotential5 so that the theory is free of fermiondoublers None of these models is supersymmetri13 under the dis13retization of any of the13ontinuum supersymmetries (2)δ(1)φx = εψx δ(1)ψx = minusε((partφ)x +Wx) δ(1)ψx = 0

δ(2)φx = ψxε δ(2)ψx = 0 δ(2)ψx = ((partφ)x minusWx)ε(9)however at least for a free theory both supersymmetries are realized in two of the modelswith antisymmetri13 matri13es (partxy) Thus we might expe13t a better approximation to the13ontinuum theory for these models In fa13t it will turn out in se13tion 3 that this behaviorpertains to the intera13ting 13ase e g the masses extra13ted from these two models are mu13h13loser to their 13ontinuum values than those from the third theory Truly supersymmetri13(improved) latti13e models will be 13onsidered in se13tion 23221 The unimproved models in detail(i) Naive latti13e model with Wilson fermionsThe most naive dis13retization is given by the a13tion (8) with an additional Wilson term5In general Wx is not equal to W prime(φx) The denition for ea13h model 13an be found below7

shifting the derivative part as explained in (7) i eS

(1)1d = Snaive part = partb Wx = W prime(φx) and Wxy = W primeprime(φx)δxy (10)The Wilson term removes fermioni13 as well as bosoni13 doublers This a13tion has no su-persymmetries at all and bosoni13 and fermioni13 ex13itations have dierent masses in the13ontinuum limit in the presen13e of intera13tions Even the free model has no exa13t super-symmetry this 13an be tra13ed ba13k to the fa13t that the derivative partb is not antisymmetri13(ii) Naive latti13e model with shifted superpotentialAn alternative way to remove the fermion doublers employed by Golterman and Pet13herand later Catterall and Gregory [10 36 is to use the (unshifted) antisymmetri13 matrix (partxy)and add a Wilson term to the superpotential

S(2)1d = Snaive part = part Wx = minusa

2(∆φ)x +W prime(φx) Wxy = minusa

2∆xy +W primeprime(φx)δxy (11)It should be noted that (as 13ompared to (10)) only the bosoni13 terms are 13hanged Thismodel is only supersymmetri13 without intera13tion i e for W prime(φx) = mφx In the inter-a13ting 13ase all susy Ward identities are violated The breaking is equally strong for bothsupersymmetries(iii) Naive latti13e model with SLAC derivativeNaively one might expe13t the supersymmetry breaking ee13t in the naive latti13e a13tion (8)with SLAC derivative (4) to be of the same magnitude as with ba13kward derivative partbSurprisingly enough this is not so it will turn out that the mass extra13tion from the model

S(3)1d = Snaive part = part slac Wx = W prime(φx) Wxy = W primeprime(φx)δxy (12)is about as good as for the improved a13tions 13onsidered below This is again related to thefa13t that the derivative is antisymmetri13 su13h that the free model with SLAC derivativeadmits both supersymmetries (in 13ontrast to the model with ba13kward derivative)23 Latti13e models with improvementIn order to preserve one of the two latti13e supersymmetries in (9) for intera13ting theoriesthe naive dis13retization (8) should be amended by extra terms whi13h turn into surfa13e termsin the 13ontinuum limit and reinstall part of the 13ontinuum supersymmetry on the latti13eSu13h invariant latti13e models may be 13onstru13ted with the help of a Ni13olai map φ 7rarr ξ(φ)8

of the bosoni13 variables In terms of the Ni13olai variables ξx(φ) the improved latti13e a13tionstake the simple formSsusy =

a

2

sum

x

ξx(φ)2 + asum

xy

ψxpartξxpartφy

ψy (13)This is a dis13retization of the most general supersymmetri13 a13tion in terms of a real bosoni13variable and two real Grayumlmann variables on the 13ir13le [27 It is easily seen to be invariantunder the rst type of transformationsδ(1)φx = εψx δ(1)ψx = minusεξx δ(1)ψx = 0 (14)For the parti13ular 13hoi13e ξx(φ) = (partφ)x +Wx the supersymmetri13 a13tion (13) be13omes

Ssusy =a

2

sum

x

((partφ)x +Wx)2 + a

sum

xy

ψx (partxy +Wxy)ψy

= Snaive + asum

x

Wx(φ) (partφ)x (15)and the supersymmetry transformation (14) is identi13al to δ(1) in (9) So the improvedmodel (15) is invariant under the rst supersymmetry δ(1) for arbitrary superpotentials Itdiers from the naive dis13retization (8) of the 13ontinuum a13tion (1) by the improvementterm asum

xWx (partφ)x whi13h turns into an integral over a total derivative in the 13ontinuumand hen13e zero for periodi13 boundary 13onditions The improvement term is needed for aninvarian13e of the latti13e a13tion under one supersymmetry transformation For an intera13tingtheory the latti13e a13tion (15) is not invariant under the other supersymmetry transforma-tion δ(2) with parameter ε an a13tion preserving only this symmetry 13an be analogously13onstru13tedSsusy =

a

2

sum

x

ξx(φ)2 + asum

xy

ψxpartξypartφx

ψy (16)It is invariant under the nilpotent supersymmetry transformationsδ(2)φx = ψxε δ(2)ψx = 0 δ(2)ψx = minusξxε (17)To generate the same fermioni13 term as in Ssusy we 13hoose ξx = minus(partφ)x + Wx with antisym-metri13 (partxy) and symmetri13 Wxy Then the supersymmetry (17) agrees with δ(2) in (9) andthe a13tion takes the form

Ssusy =a

2

sum

x

((partφ)x minusWx)2 + a

sum

xy


= Snaive minus asum

x

Wx(φ) (partφ)x (18)9

For periodi13 elds Ssusy and Ssusy 13onverge to the same 13ontinuum limit On the latti13e theyare only equal in the nonintera13ting 13ase6231 The improved models in detailWe 13onsider three supersymmetri13 versions of the dis13retization (15) with improvementterm(iv) Supersymmetri13 model with Wilson fermions and Ito pres13riptionIn order to avoid doublers and at the same time keep half of supersymmetry we use theantisymmetri13 matrix (partxy) and shift the superpotential by a Wilson term [407 The 13orre-sponding modelS

(4)1d = Ssusy part = part Wx = minusa


2∆xy +W primeprime(φx)δxy (19)is invariant under the supersymmetry δ(1) [36 Of 13ourse the non-intera13ting model is alsoinvariant under δ(2)With these denitions the improvement term is given by the well-known Ito pres13ription

sum

xWprime(φx) (φx minus φxminus1)(v) Supersymmetri13 models with Wilson fermions and Stratonovi13h pres13riptionInstead of the Ito pres13ription we 13an 13hoose the Stratonovi13h s13heme [40 for the evaluationof the surfa13e term sum

xWprime(σx) (φx minus φxminus1) with σx = 1

2(φx + φxminus1)8 The 13orrespondinga13tion 13an be obtained from (13) with a Ni13olai variable ξx(φ) = (partbφ)x +W prime(σx)

S(5)1d = Ssusy part = part Wx = minusa

2(∆φ)x +W prime(σx) Wxy = minusa

2∆xy +

partW prime(σx)

partφy (20)One should note that this pro13edure diers from the one proposed in [38 where the fermionsare rst integrated out in the 13ontinuum theory and only then a Stratonovi13h interpretationis given for the surfa13e term in this 13ase the fermioni13 path integral of the Eu13lidean evolu-tion operator has to be dened in a non-standard way in order for the bosoni13 Stratonovi13hJa13obian to 13an13el the fermion determinant6In order to preserve the se13ond supersymmetry δ(2) also for Ssusy in (15) in the absen13e of intera13tionsits denition will have to be slightly modied only for the Stratonovi13h pres13ription to be dis13ussed below7It is obvious that this is equivalent to working with a shifted latti13e derivative as in (7) and an unshiftedsuperpotential sin13e the a13tion now only depends on the invariant 13ombination ξx8For monomial superpotentialsW (φ) = φk k = 1 2 this pres13ription is equivalent to the pres13ription

sum

x12

(

W prime(φx) +W prime(φxminus1))

(φx minus φxminus1) in the latter 13ase the superpotential terms are evaluated only ata given latti13e site 10

We will see that 13ompared to the fermion determinant involving 13ontinuum derivativesthe 13ontinuum limit of the fermion determinant for the Ito pres13ription is o by a fa13tordepending on the superpotential whereas the Stratonovi13h pres13ription reprodu13es exa13tlythe desired 13ontinuum resultSin13e (20) was 13onstru13ted from (13) in terms of Ni13olai variables it is manifestly super-symmetri13 under δ(1) as given in (14) The dis13retization of the se13ond supersymmetry isin general not preserved on the latti13e not even in the free 13ase This latter fa13t suggests amodi13ation of ξx(φ) in (17) toξx(φ) = minus(partφ)x minus

a

2(∆φ)x +W prime(σprime

x) (21)This 13hanges ee13tively the ba13kward- into a forward-derivative and the derivative of thesuperpotential is evaluated now at σprimex = 1

2(φx + φx+1) With these denitions δ(2) is asymmetry of the a13tion (20) in the absen13e of intera13tions It is also this variation withwhi13h we 13ompute Ward identities in se13tion 3(vi) Supersymmetri13 models with SLAC derivativeIn order to avoid fermion doublers we 13an spe13ialize part to be the SLAC derivative

S(6)1d = Ssusy part = part slac Wx = W prime(φx) (22)In spite of its nonlo13ality the fermion and boson masses extra13ted from two-point fun13tionsprove to approa13h the 13ontinuum value quite fast the quality turns out to be 13omparableto that of the Stratonovi13h pres13ription The intera13ting supersymmetri13 model with SLACderivative is only invariant under δ(1) in (14) by 13onstru13tion24 Fermion determinantsIn this subse13tion we demonstrate whi13h sign of the Wilson term we must 13hoose in orderto guarantee positivity of the fermion determinant After that it will be13ome 13lear thatas 13ompared to the value for the 13ontinuum operator the fermion determinant for the Itopres13ription is o by a fa13tor depending on the superpotential whereas the Stratonovi13hpres13ription and the SLAC derivative reprodu13e the desired 13ontinuum result241 Sign of the determinantsFor a real fermion matrix partxy + Wxy 13omplex eigenvalues λ appear pairwise as λλ in thedeterminant Hen13e the determinant 13an only be13ome negative through real eigenvalues11

Sin13e without loss of generality eigenve13tors vx to real eigenvalues λ 13an be taken to be real(otherwise take vx + vlowastx) and normalized only the symmetri13 part of the fermion matrix13ontributes to a real λλ =

sum

xy

vx (partxy +Wxy) vy =sum

xy

vx(

part sxy +Wxy

)

vy (23)For the antisymmetri13 SLAC derivative the symmetri13 part parts is absent and no Wilson termis required Wxy = W prime(φx)δxy su13h that the real eigenvalues are given byλ =

sum

x

W primeprime(φx)v2x (24)We 13on13lude that all real eigenvalues and thus the determinant will be positive for themodels (iii) and (vi) in 13ase W primeprime is nonnegative denite For the models (i) (ii) and (iv)with Wilson term the real eigenvalues are given by

λ = minusar2

sum

xy

vx∆xyvy +sum

v

W primeprime(φx)v2x (25)Sin13e minus∆ is positive all real eigenvalues λ and therefore the determinant will be positivedenite in 13ase the W primeprime(φx) are nonnegative and r gt 0 as usual in this paper we 13hoose

r = 1 in this 13ase Vi13e versa for a negative W primeprime(φx) we would have to 13hange the sign ofthe Wilson term (i e 13hoose r = minus1) for the fermioni13 determinant to stay positive9 Inthe following subse13tion we shall prove by an expli13it 13al13ulation that also for the model (v)with Stratonovi13h pres13ription the fermioni13 determinant is positive for positive W primeprimeIf W primeprime is neither positive nor negative denite positivity of the fermion determinant is notexpe13ted In fa13t for instan13e in the parti13ular example ofW (andW primeprime) being an odd powerof φ we have to expe13t a 13hange of sign From the interpretation of the Witten indexZW =

int

perbc

DψDψDφ eminusS =

int

Dφ det(part +W primeprime(φ)) eminusSbos (26)(for the path integral with periodi13 boundary 13onditions for all elds) as the winding numberof the Ni13olai map ξ = partφ +W prime regarded as a map from the spa13e of bosoni13 variables toitself [42 we expe13t that there may be phases with broken supersymmetry forW prime even theWitten index vanishes This would be impossible for a determinant with denite sign9Either way this part of the a13tion satises site- as well as link ree13tion positivity [4112

242 Cal13ulating the determinantsThe fermioni13 determinant for the Ito and Stratonovi13h pres13ription 13an be 13omputed ex-a13tly The (regularized) determinant of the 13ontinuum operator partτ +W primeprime(φ(τ)) on a 13ir13leof radius β 13an easily be seen to be [43det

(

partτ +W primeprime(φ(τ))

partτ +m

)

=sinh (1

2

int β

0dτ W primeprime(φ(τ)))

sinh(β2m)

(27)this is the value with whi13h we have to 13ompare the latti13e results Note that for a non-negative W primeprime the determinant is positiveFor Itos 13al13ulus with Wilson fermions the ratio of the determinant of the fermion matrixfor the intera13ting theorypartxy +Wxy = partb

xy +W primeprime(φx)δxy (28)to that of the free theory is given bydet

(

part + (Wxy)

partb +m1 )

Ito

=

prod

(1 + aW primeprime(φx)) minus 1

(1 + am)N minus 1 (29)It is positive for positive W primeprime(φ) and this agrees with the results in the previous se13tion For

N = βararr infin the produ13t 13onverges to10det

(

part + (Wxy)

partb +m1 )

Ito

Nrarrinfinminusrarr eR β0dτ W primeprime(φ(τ)) dτ2

e1

2βm2

det

(


partτ +m

) (30)sin13e lnprod

x(1 + aW primeprime(φx)) =sum

ln(1 + aW primeprime(φx)) rarrint

dxW primeprime(φx) The limit (30) diers bya eld dependent fa13tor from the 13ontinuum resultFor Wilson fermions with Stratonovi13h pres13ription the regularized determinant of thefermion matrixpartxy +Wxy = partb

xy +1

2W primeprime(σx) (δxy + δxminus1y) (31)is again positive for positive W primeprime But in 13ontrast to the determinant (29) with Ito pres13rip-tion it 13onverges to the 13ontinuum result

det

(

part + (Wxy)

partb + (mxy)

)

Strat

=

prod

(1 + a2Mx) minus

prod

(1 minus a2Mx)

prod

(1 + a2m) minus prod

(1 minus a2m)

Nrarrinfinminusrarr det

(


partτ +m

)

(32)where mxy = 12(δxy + δxminus1y) and Mx = W primeprime(σx) One 13an show that for N rarr infin thefermioni13 determinant with SLAC derivative 13onverges rapidly to the 13ontinuum result10In a 13ompletely analogous manner the right-derivative would lead to the inverse prefa13tor in front ofthe 13ontinuum result 13

3 Simulations of supersymmetri13 quantum me13hani13sWe have performed high pre13ision Monte-Carlo simulations to investigate the quality ofthe six latti13e approximations introdu13ed in subse13tions 221 and 231 The models witha13tions S(1)1d S

(2)1d and S

(3)1d are not supersymmetri13 whereas the a13tions S(4)

1d S(5)1d and S

(6)1dpreserve the supersymmetry δ(1)31 Ee13tive Masses on the latti13eIn order to determine the masses we have 13al13ulated the fermioni13 and bosoni13 two pointfun13tions

G(n)bos(x) = 〈φxφ0〉 and G

(n)ferm(x) = 〈ψxψ0〉 (33)in all models (n) and tted their logarithms with a linear fun13tion This way of determiningthe masses mbos(a) and mferm(a) from the slope of the linear t works well for all modelswith ultralo13al derivatives Details on te13hni13al aspe13ts of the extra13tion of ee13tive masses13an be found in se13tion 53311 Models without intera13tionThe a13tions of the non-intera13ting models are quadrati13 in the eld variables

Sfree =1

2

sum

xy

φxKxyφy +sum

xy

ψxMxyψy (34)A13tually for W prime(φ) = mφ all the improvement terms vanish and only four of the six a13tionsintrodu13ed in the last se13tion are dierent The 13orresponding matri13esM and K are givenin the following list11S

(1)free S

(4)free = S

(2)free S

(5)free S

(6)free = S

(3)free

M partb +m partb +m (1 minus am2

)partb +m part slac +m

K minus∆ +m2 minus∆ +m2 minus am∆ minus(

1 minus (am2

)2)

∆ +m2 minus(part slac)2 +m2

= MTM = MTM = MTM

(35)For the a13tions in the last three 13olumns we have detK = (detM)2 as required by su-persymmetry This is not true for the non-supersymmetri13 naive model S(1)free with Wilsonfermions11Note that partb + (partb)T = minus∆ and minuspart2 + 1

4a2∆2 = minus∆14

Without intera13tions the masses of all models as determined by numeri13ally inverting Mand K 13onverge to the same 13ontinuum limit m For the free supersymmetri13 models (ii)-(vi) mbos(a) and mferm(a) roughly 13oin13ide even for nite latti13e spa13ings This is to beexpe13ted for supersymmetri13 theories In the rst model mbos(a) for nite a is already very13lose to its 13ontinuum value in 13ontrast to mferm(a) whi13h for nite a is 13onsiderably smallerthan mbos(a) Sin13e the free supersymmetri13 a13tion S(4)free has the same fermioni13 mass as

S(1)free and sin13embos(a) asymp mferm(a) for this model we 13on13lude that its bosoni13 mass for nitea is notably smaller than its value in the 13ontinuum limit Thus when supersymmetrizingthe naive model with Wilson fermions we pay a pri13e the boson masses get worse whileapproa13hing the fermion massesThe situation is mu13h better for the other supersymmetri13 models with SLAC derivativeor Wilson fermions with Stratonovi13h pres13ription The masses are equal and very 13lose tothe 13ontinuum result already for nite a The masses for the Stratonovi13h pres13ription are13omparable to the boson masses of the naive model without supersymmetry The massesfor the free models and their dependen13e on a are depi13ted in gure 1

PSfrag repla13ements

latti13e spa13ing aee13tivemass

m

m(1)bos

m(1)ferm

m(5)ferm

m(6)ferm

00102030405060708091

0 0005 001 0015 002 0025

001020304050607080918899294969810

102Figure 1 The masses determined by numeri13ally inverting the kineti13 operators for the freetheories There are only 4 dierent masses 13f (35)15



m

naive SLAC fermions

naive Wilson bosonsnaive Wilson fermionsnaive SLAC bosonsnaive SLAC fermions

00102030405060708091

0005 001 0015 002 0025

0010203040506070809112131415161718

Figure 2 Boson and fermion masses for Wilson fermions without improvement (model (i))and non-supersymmetri13 SLAC fermions (model (iii)) The parameters for the linear ts13an be found in table 1From (35) we 13on13lude thatG

(4)ferm(xm) =

(

1 minus am

2

)

G(5)ferm

(

xm(

1 minus am

2

))

(36)The mass m(5)ferm(a) extra13ted from G

(5)ferm is 13lose to the 13ontinuum value m su13h that

m(124)ferm (a) = m

(24)bos (a) asymp m

(

1 minus am

2

)

(37)and this simple relation explains why the linear t through the masses m(1)ferm(a) markedwith red dots in gure 1 has su13h a large negative slope312 Models with intera13tionWe have 13al13ulated the masses for the intera13ting models with even superpotential

W (φ) =m

2φ2 +

g

4φ4 =rArr W prime(φ)2 = m2φ2 + 2mgφ4 + g2φ6 (38)Sin13e in the weak 13oupling regime the results are 13omparable to those of the free models wehave simulated the models at strong 13oupling In order to 13ompare our results with those16

of Catterall and Gregory in [36 we have pi13ked their values m = 10 and g = 100 for whi13hthe dimensionless ratio gm2 equals unity The energy of the lowest ex13ited state has been13al13ulated by diagonalizing the Hamiltonian on large latti13es with small a and alternativelywith the shooting method With both methods we obtain the 13ontinuum valuemphys = 16865 (39)We now summarize the results of our MC simulations As in the free 13ase mbos(a) 6=

mferm(a) for the non-supersymmetri13 model with a13tion S(1)1d see gure 2 In addition for

g 6= 0 their 13ontinuum values are dierent and none of the two values agrees with (39) Thishas been predi13ted earlier by Giedt et al [37 We 13on13lude that the naive latti13e modelwith Wilson fermions is not supersymmetri13 for ararr 0The Monte-Carlo results are mu13h better for the se13ond model with a13tion S(2)1d as givenin (11) Although this model is not supersymmetri13 its boson and fermion masses arealmost equal for nite latti13e spa13ings Linear extrapolations to vanishing a yield m(2)

bos(0) =

1668 plusmn 005 and m(2)ferm(0) = 1673 plusmn 004 whi13h are quite 13lose to the 13orre13t value 16865The results for the non-supersymmetri13 model with a13tion S

(2)1d and the supersymmetri13model with a13tion S(4)

1d are almost identi13al similarly as for the free models The massesfor various latti13e 13onstants between 0005 and 003 for the two models are depi13ted ingure 3 The 13orre13tions to the 13ontinuum value are of order O(a) and are as big as for the13orresponding free models The slope and inter13epts for the linear ts are listed in table 1At nite latti13e spa13ing a the masses mbosferm(a) for model (v) with Wilson fermions andStratonovi13h pres13ription are mu13h 13loser to their respe13tive 13ontinuum limits than forthe model (iv) with Ito pres13ription Furthermore the extrapolated 13ontinuum massesm

(5)bos(0) = 1678 plusmn 004 and m

(5)ferm(0) = 1677 plusmn 002 are very 13lose to the 13orre13t value(39) The data points for the supersymmetri13 models with Wilson fermions are depi13ted ingure 4 Again the slope and inter13epts of the linear ts 13an be found in table 1 Of alllatti13e models with ultralo13al derivatives 13onsidered in this paper this model yields the bestpredi13tionsThe model with Stratonovi13h pres13ription for the improvement term is outperformed onlyby the models (iii) and (vi) with nonlo13al SLAC derivative for fermions and bosons Thisobservation is not surprising sin13e the remarkably high numeri13al pre13ision of supersymmet-ri13 latti13e models with SLAC derivative has been demonstrated earlier in the Hamiltonianapproa13h in [9 Furthermore this is in line with our results for the free models see gure 1The masses for the intera13ting unimproved model (iii) are plotted in gure 2 and those17



m

shifted superpot fermions

improved Wilson bosonsimproved Wilson fermionsshifted superpot bosonsshifted superpot fermions

00102030405060708091

0005 001 0015 002 0025 003

001020304050607080911212513135141451515516165

Figure 3 The masses mbos(a) and mferm(a) for models (ii) (shifted superpotential) and (iv)(Ito improvement)



m

Wilson fermions Strat

Wilson bosons ItoWilson fermions ItoWilson bosons StratWilson fermions Strat

00102030405060708091

0005 001 0015 002 0025 003

00102030405060708091121251313514145151551616517

Figure 4 Masses for supersymmetri13 models (vi) (Ito pres13ription) and (v) (Stratonovi13hpres13ription) Only latti13es with at least 61 sites are in13luded in the t18



m

improved SLAC fermions

Wilson bosons ItoWilson fermions Itoimproved SLAC bosonsimproved SLAC fermions

00102030405060708091

0005 001 0015 002 0025 003

00102030405060708091121251313514145151551616517

Figure 5 Masses for the supersymmetri13 model (vi) with SLAC derivative as 13ompared tothe masses of the model (vi) with Wilson fermions and Ito pres13riptionfor the improved model (vi) in gure 5 Even for moderate latti13e spa13ings the massesmbos(a) asymp mferm(a) are very 13lose to their 13ontinuum limits m(6)

bos(0) = 1684 plusmn 003 andm

(6)ferm(0) = 1681 plusmn 001 whi13h in turn are o the true value 16865 by only some tenthof a per13ent The extrapolated masses for the unimproved latti13e model in table 1 have a13omparable pre13ision But of 13ourse there is no free lun13h sin13e for the SLAC derivativeone must smooth the two-point fun13tions Gbosferm(x) with a suitable lter for a sensiblemass extra13tion Details on the ltering 13an be found in subse13tion 53In the following table we list the slopes k and inter13epts m(0) of the linear ts

mbos(a) = kbos middot a+mbos(0) and mferm(a) = kferm middot a+mferm(0) (40)to the measured masses for the six latti13e models 13onsideredThe linear ts for the models with improvement (iii)-(vi) are 13ompared in gure 6 Latti13esupersymmetry guarantees that the boson and fermion masses are equal for these models19

model kbos mbos(0) kferm mferm(0)

S(1)1d 13952 plusmn 845 1223 plusmn 008 minus18625 plusmn 498 1840 plusmn 005

S(2)1d minus13685 plusmn 522 1668 plusmn 005 minus14610 plusmn 384 1673 plusmn 004




S(6)1d minus1797 plusmn 241 1684 plusmn 003 minus1853 plusmn 091 1681 plusmn 001Table 1 Slope and inter13epts of linear interpolations for the masses



m

13ontinuum value 16865

improved SLAC ferm

Wilson bos ItoWilson ferm ItoWilson bos StratWilson ferm Stratimproved SLAC bosimproved SLAC ferm

00102030405060708091

0 0005 001 0015 002 0025 003 0035

00102030405060708091121251313514145151551616517

Figure 6 Linear ts to the masses of all three supersymmetri13 latti13e models32 Ward identitiesThe invarian13e of the path integral measure under supersymmetry transformations leads toa set of Ward identities 13onne13ting bosoni13 and fermioni13 13orrelation fun13tions Namelythe generating fun13tional for Greens fun13tions should be invariant under supersymmetryvariations of the elds0 = δZ[J θ θ] =

int

D(φ ψ) eminusS+P

x(Jxφx+θxψx+θxψx)(

sum

y

(Jyδφy+θyδψy+θyδψy)minusδS)

(41)20


latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2 Ward-Identity 1 m = 10 g = 0Ward-Identity 2 m = 10 g = 0

0 5 10 15 20-0002-0001000010002

Figure 7 Ward identities for the free theory model (ii) (with shifted superpotential)Ward identities are obtained from derivatives of this equation with respe13t to the sour13esThe se13ond derivative part2partJxpartθy leads to the S13hwinger-Dyson equation〈φxψy δ(1)S〉 = 〈φx δ(1)ψy〉 + 〈ψyδ(1)φx〉 (42)whereas part2partJxpartθy yields〈φxψy δ(2)S〉 = 〈φx δ(2)ψy〉 + 〈ψyδ(2)φx〉 (43)For the improved models (iv)(vi) in subse13tion 231 the a13tions are manifestly invariantunder δ(1) and the left-hand side of (42) vanishes Thus for these models the following Wardidentities hold on the latti13e

〈ψxψy〉 minus 〈φxξy〉 = 0 (44)This 13an be 13onrmed in numeri13al 13he13ks and merely serves as a test bed for the pre13ision ofthe algorithms The dis13retization of the se13ond 13ontinuum supersymmetry transformationhowever only leaves these latti13e a13tions invariant in the free 13ase With intera13tions theterm δ(2)S leads to a nonvanishing left-hand side in the S13hwinger-Dyson identities (43)whi13h therefore measures the amount by whi13h the se13ond 13ontinuum supersymmetry isbroken by the dis13retization Sin13e this supersymmetry 13an be made manifest by 13hoosing21


latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2

Ward-Identity 1 m = 10 g = 800Ward-Identity 2 m = 10 g = 800

00102030405060708091

0 5 10 15 20

00102030405060708090001-0002-0001000010002

Figure 8 Ward identities for the theory at strong 13oupling model (ii) (with shifted super-potential)dierent Ni13olai variables ξ the supersymmetry breaking terms are the dieren13e of botha13tions (15) and (18) i e the dis13retization of the surfa13e terms (for (iv) and (vi))For the a13tions in (i)(iii) all supersymmetries are broken by the dis13retization the 13or-responding S13hwinger-Dyson equations again 13an serve as a measure for the quality of thelatti13e approximation to supersymmetry Barring intera13tions we expe13t them to hold forboth supersymmetry transformations in (ii) and (iii)321 Ward identities of the unimproved modelsThe Ward identities have been simulated for models with the same superpotential as in (38)Sin13e at weak 13oupling similar results 13an be expe13ted as in the free theory with g = 0 wehave also studied the theory at strong 13oupling with g = 800 and m = 10 13orresponding toa dimensionless ratio gm2 = 8The naive dis13retization (i) is not supersymmetri13 even without intera13tions therefore we13on13entrate on the other models where the supersymmetry of the free theory sets a s13ale forthe quality of the simulation of the broken supersymmetries in the intera13ting 13ase For theunimproved models we 13an use translation invarian13e and measure the right-hand sides of22


latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708090001-0006-0004-00020000200040006

Figure 9 Ward identities for the free theory model (iv) (Wilson fermions with the Itopres13ription)the Ward identities (42) and (43) i eR(1)xminusy = 〈ψxψy〉 minus 〈φx(partφ)y〉 minus 〈φxWy〉 (45)and

R(2)xminusy = 〈φx (partφ)y〉 minus 〈φxWy〉 minus 〈ψxψy〉 (46)as fun13tions of xminus y It should be noted that without intera13tions the rst Ward identityredu13es to a matrix identity Dminus1minus (DTD)minus1DT = 0 for the free Dira13 operator Dxy = partxy+

mδxy if one uses that 〈ψxψy〉 = Dminus1xy and 〈φxφy〉 = (DTD)minus1

xy The 13orresponding data forthe free theory in the 13ase of the model with shifted superpotential (model (ii)) are shown ingure 7 In order to keep statisti13al errors small in all situations 4 runs with 106 independent13ongurations were evaluated Within our numeri13al pre13ision supersymmetry is broken forthis model if the simulation data in the intera13ting 13ase ex13eeds the bounds set by the freetheory The results for (45) and (46) at g = 800 are displayed in gure 8 Remarkably thestatisti13al error is mu13h smaller than in the free theory supersymmetry breaking is equallystrong for both supersymmetries δ(1) and δ(2) from (9) at strong 13oupling23

322 Ward identities of the improved modelsFor Wilson fermions with exa13t supersymmetry and the Ito pres13ription (model (iv)) thestatisti13al error as measured by the free Ward identities is roughly of the same size as for theunimproved model (ii) 13f gure 9 Again this was obtained by 4 runs with 106 independent13ongurations As expe13ted the rst supersymmetry δ(1) (13f (14) and gure 10) is preservedeven at strong 13oupling however the supersymmetry breaking ee13ts for δ(2) are about threetimes as large as for the 13orresponding symmetry in the unimproved situation


latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708090001-0006-0004-00020000200040006

Figure 10 Ward identities for the theory at strong 13oupling model (iv) (Wilson fermionswith the Ito pres13ription)For the free supersymmetri13 model with Stratonovi13h pres13ription (model (iv)) the se13ondsupersymmetry (17) is only a symmetry with the denition (21) of ξ The Ward identitiesof both supersymmetry transformations〈ψxψy〉 minus 〈φx(partφ)y〉 minus 〈φxW prime(φx+φxminus1

2)〉 = 0

〈φx (partφ)y〉 minus 〈φxW prime(φx+φx+1

2)〉 minus 〈ψxψy〉 = 0 (47)redu13e in the free theory with W prime(φ) = mφ to matrix identities for the free Dira13 operator

Dxy = partxy + m2(δxy + δxyminus1)

Dminus1 minus (DTD)minus1DT = 024


latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708091-002-0015-001-0005000050010015002

Figure 11 Ward identities for the free theory model (v) (Wilson fermions with theStratonovit13h pres13ription)


latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708091-002-0015-001-0005000050010015002

Figure 12 Ward identities for the theory at strong 13oupling model (v) (Wilson fermionswith the Stratonovit13h pres13ription) 25

minus(DTD)minus1D + (Dminus1)T = 0 (48)Here the se13ond identity holds sin13e D is 13ir13ulant and therefore normal The 13orrespondingplot of the left-hand sides is shown in gure 11 This determines the mean error abovewhi13h we take supersymmetry to be broken if we swit13h on intera13tions In gure 12the rst supersymmetry is preserved within a high numeri13al a1313ura13y whereas the se13ondsupersymmetry is 13learly broken by ee13ts about three times the size of the supersymmetryviolation in the model with Ito pres13riptionFor the supersymmetri13 model with SLAC derivative the Ward identities are satised withinstatisti13al error bounds but determining the mean error in an analogous manner fails sin13ethe 13ontributions of the bosoni13 two-point fun13tions 〈φx(partφ)y〉 in the free theory lead tolarge errors whi13h obs13ure the interpretation of the 13orresponding Ward identities At


latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708091-0025-002-0015-001-00050000500100150020025

Figure 13 Ward identities for the theory at strong 13oupling model (vi) (supersymmetri13model with SLAC fermions)strong 13oupling however the intera13tion terms dominate the a13tion and the u13tuationsof the bosoni13 propagators be13ome in13reasingly less important so that we obtain ratherpre13ise results at g = 800 Within the error bounds of model (v) the rst supersymmetryis obviously preserved in the intera13ting theory (13f g 13) whereas the supersymmetrybreaking ee13ts of the se13ond supersymmetry are about as large as for Wilson fermions with26

the Stratonovi13h pres13ription4 The N = 2 Wess-Zumino model in 2 dimensionsIn this se13tion we study dierent dis13retizations of the two-dimensional Wess-Zumino modelwith (2 2) supersymmetry The minimal variant 13ontains a Dira13 spinor eld and two reals13alar elds ϕa whi13h are 13ombined into a 13omplex s13alar eld φ = ϕ1 + iϕ2 Also we usethe 13omplex 13oordinate z = x1+ix2 and its 13omplex 13onjugate z in Eu13lidean spa13etime anddenote the 13orresponding derivatives by part = 12(part1 minus ipart2) and part respe13tively The Eu13lideana13tion 13ontains the rst and se13ond derivatives W prime and W primeprime of a holomorphi13 superpotential

W (φ) with respe13t to the 13omplex eld φScont =

int

d2x(

2partφpartφ +1

2|W prime|2 + ψMψ

)

M = part +W primeprimeP+ + W primeprimePminus (49)where Pplusmn = 12(1plusmn γ3) are the 13hiral proje13tors In the Weyl basis with γ1 = σ1 γ2 = minusσ2and γ3 = iγ1γ2 = σ3 the 13omplex spinors 13an be de13omposed a1313ording to

ψ =

(

ψ1

ψ2

) and ψ = (ψ1 ψ2) (50)In this basis the supersymmetry transformations leaving Scont invariant areδφ = ψ1ε1 + ε1ψ

1 δψ1 = minus12W primeε1 minus partφε2 δψ1 = minus1

2W primeε1 + partφε2

δφ = ψ2ε2 + ε2ψ2 δψ2 = minuspartφε1 minus 1

2W primeε2 δψ2 = partφε1 minus 1

2W primeε2

(51)Similarly as in quantum me13hani13s a naive dis13retization of this model breaks all four super-symmetries In order to keep one supersymmetry one 13an add an improvement term In whatfollows we shall only 13onsider improved models they dier by our 13hoi13e of the latti13e deriva-tives Instead of trying to generalize the Stratonovi13h pres13ription to the two-dimensionalsituation we nd in subse13tion 41 that a non-standard 13hoi13e of the Wilson term leads toan improved behavior in the limit of vanishing latti13e spa13ing This is 13orroborated by theresults of our simulations for the 13ase of a 13ubi13 superpotential W = 12mφ2 + 1

3gφ3 whi13hwe present in subse13tion 4341 Latti13e models with improvementWe start with a two-dimensional periodi13 N1 timesN2 latti13e Λ with 13omplex bosoni13 variables

φx and two 13omplex spinors ψx ψx on the latti13e sites x = (x1 x2) isin Λ Again two dierent27

antisymmetri13 latti13e derivatives in dire13tion micro are used These are the ultralo13al derivativepartmicroxy =

1

2a

(

δx+emicroy minus δxminusemicroy

) (52)with doublers and the nonlo13al SLAC derivative without doublers whi13h for odd N1 N2readspart slac

1xy = part slacx1 6=y1δx2y2 part slac

2xy = part slacx2 6=y2δx1y1 and part slac

microxx = 0 (53)(analogously to the one-dimensional SLAC derivative dened in (4)) Later we shall removethe fermioni13 doublers of γmicropartmicro by introdu13ing two types of Wilson terms both 13ontainingthe latti13e Lapla13ian ∆ dened by(∆φ)(x) =

1

a2

(

sum

micro=12

[φ(x+ emicro) + φ(xminus emicro)] minus 4φ(x))

(54)As for the 13ontinuum model we use the holomorphi13 latti13e derivative partxy = 12(part1xyminusipart2xy)The Ni13olai variables of one-dimensional systems are easily generalized to two dimensions

ξx = 2(partφ)x +Wx ξx = 2(partφ)x + Wx (55)again Wx denotes terms (to be spe13ied below) derived from the superpotential Thebosoni13 part of the a13tion is Gaussian in these variables Sbos = a2

2

sum

x ξxξx and has theexpli13it formSbos = a2

sum

x

(

2(partφ)x(partφ)x +Wx(partφ)x + W primex(partφ)x +

1

2|Wx|2

)

(56)For antisymmetri13 derivatives partmicro the kineti13 term has the standard form sum2amicro=1(partmicroϕa)

2x interms of the real elds ϕa in parti13ular this holds true for the ultralo13al derivative partmicro andthe SLAC derivative part slac

micro introdu13ed above The se13ond and third term in Sbos are absent inthe 13ontinuum a13tion In a naive dis13retization of the 13ontinuum model these improvementterms do not show up In the 13ontinuum limit they be13ome surfa13e terms and 13ould bedropped On the latti13e they are needed to keep one of the four supersymmetries inta13tSupersymmetry requires an additional fermioni13 term Sferm = a2sum

ψxMxyψy for a two-13omponent Dira13 spinor eld in su13h a way that the determinant of the Ja13obian matrix(

partξxpartφy partξxpartφy


)

=

(

Wxy 2partxy

2partxy Wxy

)

Wxy =partWx

partφy (57)28

for the 13hange of bosoni13 variables (φ φ) 7rarr (ξ ξ) 13an13els the fermion determinant detM A13tually the fermioni13 operatorM in (49) with γ-matri13es in the Weyl basis and 13ontinuumderivatives repla13ed by latti13e derivatives is identi13al to the Ja13obian matrix Hen13e we13hoose as the fermioni13 part of the a13tionSferm = a2

sum

xy

ψxMxyψy Mxy = γmicropartmicroxy +WxyP+ + WxyPminus

= M0 +W primeprime(φx)δxy P+ + W primeprime(φx)δxy Pminus (58)By 13onstru13tion the a13tion Ssusy = Sbos + Sferm with improvement terms is invariant underthe supersymmetry transformations generated by δ(1)δ(1)φx = εψ1

x δ(1)ψ1x = minus1

2εξx δ(1)ψ1

x = 0

δ(1)φx = εψ2x δ(1)ψ2

x = minus12εξx δ(1)ψ2

x = 0(59)this 13orresponds to a dis13retization of the 13ontinuum symmetry (51) with ε1 = ε2 = 0 and

ε1 = ε2 = ε The other three 13ontinuum supersymmetries are broken for appropriately13hosen Ni13olai variables ξ an a13tion preserving any one of the three other supersymmetries13an be 13onstru13ted analogously [3012 In this paper we are going to use the Ni13olai variable(55) and 13onsider several possibilities to remove fermion doublers42 The latti13e models in detailWe introdu13e three dierent latti13e approximations to the 13ontinuum Wess-Zumino model(49) They are all equipped with an improvement term and thus admit one supersymmetryThe rst two models 13ontain Wilson fermions and the third the SLAC derivative It willturn out that the dis13retization errors of the eigenvalues of the bosoni13 and fermioni13 kineti13operators in the free 13ase indi13ate how good the approximation to the 13ontinuum theory iswhen we turn on intera13tions(i) Supersymmetri13 model with standard Wilson termHere we 13hoose ultralo13al derivative partmicro and add a standardWilson term to the superpotentialto get rid of the doublers of γmicropartmicro so thatS(1) = Sbos + Sferm with partmicro = partmicro Wx = minusar

2(∆φ)x +W prime(φx) (60)12The 13orresponding Ni13olai variables 13an be read o from the right-hand sides of δψa in (51) for ε1 =

plusmnε2 = ε and ε = 0 or from the right-hand sides of δψa for ε1 = plusmnε2 = ε and ε = 029

For later 13onvenien13e we do not x the Wilson parameter r in this se13tion In this latti13emodel the Dira13 operator M0 in (58) takes the formM

(1)0 = γmicropartmicro minus

ar

2∆ (61)and we easily re13ognize the standard Dira13 operator for Wilson fermions The bosoni13 partof the a13tion may be expanded as

S(1)bos =

a2

2

sum

x

(

φx(Kφ)x + |W prime(φx)|2)

+ a2sum

x

(

W prime(φx) (partφminus ar

4∆φ)x + 1313) (62)with the kineti13 operator

K = minus∆ + (12ar∆)2 where ∆ = minuspartmicropartmicro (63)The last term in (62) is the improvement term a dis13retization of a surfa13e term inthe 13ontinuum theory Note that even for the free massive model with W prime(φ) = mφ theimprovement term minus1

2(amr) a2

sum

φx(∆φ)x is non-zero su13h that (62) be13omesS

(1)bos =

a2

2

sum

xy

φxK(1)xy φy with K(1) = K +m2 minus arm∆ (64)The eigenvalues of the 13ommuting operators minus∆ and minus∆ are p2 and p2 where

pmicro =1

asin apmicro pmicro =

2

asin

(apmicro2

) with pmicro =2πkmicroNmicro

kmicro isin 1 2 Nmicro (65)su13h that the eigenvalues of the matrix K(1) in (64) aremicrop =

p2 + (m+ 12arp2)2 (66)On the other hand the eigenvalues of the free Dira13 operator M (1)

0 +m are given byλplusmnp = m+ 1

2arp2 plusmn i|

p| (67)Thus microp = λ+p λ

minusp and the fermioni13 and bosoni13 determinants 13oin13ide for the free theoryThe dis13retization errors for these eigenvalues are of order a

microp = p2 +m2 + (arm)p2 +O(a2) λplusmnp = plusmni|p| +m+ 12ar p2 +O(a2) (68)It is remarkable that the bosoni13 part of the a13tion in the 13ontinuum is an even fun13tion ofthe mass whereas its dis13retization (62) is not This spoils the sign freedom in the fermion30

mass term on the latti13e and motivates the following se13ond possibility(ii) Supersymmetri13 model with non-standard Wilson termAgain we 13hoose the ultralo13al derivatives partmicro but now add a non-standard Wilson term tothe superpotential to get rid of the doublers so that the a13tion now isS(2) = Sbos + Sferm with partmicro = partmicro Wx =

iar

2(∆φ)x +W prime(φx) (69)This 13hoi13e is equivalent to an imaginary value of the Wilson parameter inside the holomor-phi13 superpotential Now the Dira13 operator M0 in (58) has the form

M(2)0 = γmicropartmicro +

iar

2γ3∆ (70)and 13ontains a non-standard Wilson term reminis13ent of a momentum dependent twistedmass It should be noted that the only dieren13e between the bosoni13 a13tions (62) and

S(2)bos =

a2

2

sum

x

(


+ a2sum

x

(

W prime(φx) (partφminus iar

4∆φ)x + 1313) (71)is the improvement term The modied Wilson term in (69) yields an a13tion whi13h is evenin the mass m A13tually for the free massive model with W prime(φ) = mφ the improvementterm vanishes and

S(2)bos =

a2

2

sum

xy

φxK(2)xy φy with K(2) = K +m2 (72)The eigenvalues of K(2) and of the free Dira13 operator M (2) +m in this 13ase are given by

microp = m2 +

p2 +(

12ar p2

)2 and λplusmnp = mplusmn i

radic

p2 +(

12arp2

)2 (73)Again the determinants of fermioni13 and bosoni13 operators are equal The added advantagein this situation is however that the Wilson parameter 13an be tuned in su13h a way thatthe dis13retization errors of the 13ontinuum eigenvalues are only of order O(a4) Namely forsmall latti13e spa13ing

microp = m2 + p2 + κ+O(a4) λplusmnp = mplusmn i(p2 + κ)12 +O(a4) (74)where the O(a2)-term κ = a2(3r2 minus 4)12sum

p4micro vanishes for 4r2 = 3 In fa13t we will see inse13tion 43 that the value

r2 =4

3(75)31

leads to the best 13ontinuum approximation and this in spite of the fa13t that the 13hoi13e (75)violates ree13tion positivity (as does the improvement term in all supersymmetri13 models)(iii) Supersymmetri13 model with SLAC derivativeThe Dira13 operator γmicropart slacmicro with nonlo13al and antisymmetri13 SLAC derivatives denedin (53) has no doublers and no Wilson terms are required this leads to the a13tion

S(3) = Sbos + Sferm with partmicro = part slacmicro Wx = W prime(φx) (76)with Dira13 operator

M(3)0 = γmicropart slac

micro (77)13f (58) For the free massive model with SLAC derivative the improvement term vanishesin parti13ular the bosoni13 part of S(3) isS

(3)bos =

a2

2

sum

xy

φxK(3)xy φy with K(3) = minus∆slac +m2 (78)all supersymmetries are realized We shall see in se13tion 43 that S(3) is a very good ap-proximation to the 13ontinuum model and in se13tion 6 that the latti13e model based on theSLAC derivative is one-loop renormalizable in spite of its nonlo13ality43 Simulations of the Wess-Zumino modelIn subse13tion 41 we have formulated the model in a 13omplex basis whi13h is natural and13onvenient for models with two supersymmetries (in parti13ular the simplest form of theNi13olai map (55) is in terms of the 13omplex s13alar elds φ = ϕ1 + iϕ2 and ξ = ξ1 + iξ2)On the other hand for numeri13al simulations it is 13onvenient to have a formulation of themodel in terms of the real 13omponents ϕa and ξa whi13h are 13ombined to real doublets

ϕ =

(

ϕ1

ϕ2

) and ξ =

(

ξ1

ξ2

)

(79)As to the fermions it is most appropriate to use a Majorana representation with real γ-matri13es γ1 = σ3 γ2 = σ1 su13h that γ3 = iγ1γ2 = minusσ2 All simulations were done for themodel with 13ubi13 superpotential W = 1

2mφ2 + 1

3gφ3 with derivative

W prime(φ) = mϕ1 + u+ i(mϕ2 + v) (80)32

where we have introdu13ed the abbreviations u = g(ϕ21 minus ϕ2

2) and v = 2gϕ1ϕ2 The a13tions13ontain a quarti13 potentialV (ϕ) = |W prime(φ)|2 = (m2 + g2ϕ2 + 2mgϕ1) ϕ2 (81)In terms of real elds the Ni13olai map (55) takes the formξ(n) = (M

(n)0 +m)ϕ +

(

u

v

) for n = 1 2 3 (82)with model-dependent free massless Dira13 operators M (n)0 as given in (61) (70) and (77)The bosoni13 a13tions for the models with standard Wilson modied Wilson and SLACfermions 13an now be written as

S(n)bos =

a2

2

sum

xy

(ϕx K(n)xy ϕy) +

a2

2

sum

x

V (ϕ) + ∆(n) (83)with model-dependent kineti13 operators K(n) as introdu13ed in (64) (72) and (78) In termsof the divergen13e and 13url of a ve13tor eld in two dimensions div ϕ = part1ϕ1 + part2ϕ2 and13url ϕ = part1ϕ2 minus part2ϕ1 the model-dependent improvement terms∆(1) = a2

sum

x

ux

(

(div ϕ)x minusar

2(∆ϕ1)x

)

minus a2sum

x

vx

(

(13url ϕ)x +ar

2(∆ϕ2)x

)

∆(2) = a2sum

x

ux

(

(div ϕ)x minusar

2(∆ϕ2)x

)

+ a2sum

x

vx

(

minus(13url ϕ)x +ar

2(∆ϕ1)x

)

(84)∆(3) = a2

sum

x

ux(div ϕ)x minus a2sum

x

vx(13url ϕ)xagain are dis13retizations of 13ontinuum surfa13e terms431 Models without intera13tionAs expe13ted the masses of all models without intera13tions 13onverge to the same 13ontinuumlimit m Sin13e all free models are supersymmetri13 (w r t two supersymmetries) bosonand fermion masses extra13ted from the two-point fun13tions 13oin13ide even at nite latti13espa13ing The masses m(a) for the models (ii) and (iii) with non-standard Wilson term(with r2 = 43) and the SLAC derivative respe13tively at nite a are already very 13lose totheir 13ontinuum limits the ee13tive mass as a fun13tion of a for the model with standardWilson term (with r = 1) has a mu13h larger slope13 This is in line with the approximationof the eigenvalues (68) and (74) to those of the 13ontinuum kineti13 operators13This behavior is reminis13ent of the masses m(a) for the 13orresponding quantum me13hani13al model inse13tion (311) 33

432 Models with intera13tionWe have 13al13ulated the masses for the intera13ting models with 13ubi13 superpotentialW (φ) =

1

2mφ2 +

1

3gφ3 (85)for masses m = 10 and 13ouplings g ranging between 0 and 1 The ee13tive masses atdierent values of the latti13e spa13ing are determined as des13ribed in se13tion 53 Again dueto supersymmetry boson and fermion masses 13oin13ide also for nite latti13e spa13ing In all13ases they 13onverge to a 13ontinuum value whi13h however 13annot reliably be distinguishedfrom the value mfree(a = 0) = 10 within error bounds This is to be expe13ted sin13e in13ontinuum perturbation theory the one-loop 13orre13ted mass is

m1minusloop = m(

1 minus g2

4πm2

) (86)in a renormalization s13heme without wave-fun13tion renormalization (this 13orresponds to a13orre13tion less than about 03 with our values of m and g) Thus signi13ant ee13ts shouldonly be seen at larger values of gm Unfortunately our simulations require reweightingswhi13h lead to rather large error bounds (whi13h grow as the 13oupling in13reases) The 13onver-gen13e behavior to the expe13ted 13ontinuum value is model-dependent The model with thenon-standard Wilson term shows the expe13ted improved behavior leading to good estimatesfor the 13ontinuum mass already at nite latti13e sizes As in quantum me13hani13s this appliesalso to the SLAC derivative5 Algorithmi13 aspe13tsIn this se13tion we briey outline the methods and algorithms we have used in our simulationsIn parti13ular we have modied the treatment of the fermion determinant in the well knownHybrid Monte Carlo algorithm (HMC) [44Although due to the low dimensionality of the models under 13onsideration the numeri13alstudies to be 13arried out are mu13h less demanding than e g four-dimensional latti13e QCDwe have to fa13e similar problems with respe13t to the treatment of the fermion elds Howeverthe 13omputational tasks at hand allow for strategies whi13h are more a1313urate and easier toimplement than what is widely used there Sin13e our models do not 13ontain any gauge34

Latti13e spa13ing Wilson Twisted WilsonmB mF mB mF01250 634(4) 64872(1) 951(8) 995(7)00833 732(3) 72730(2) 100(1) 100182(1)00625 781(8) 7768(1) 104(1) 999(1)00500 816(4) 807(1) 982(1) 993(3)Latti13e spa13ing SLAC

mB mF00769 99(2) 100(2)00667 100(1) 100(1)00526 100(1) 1000(5)00400 995(6) 999(2)00323 1003(3) 998(1)00213 983(3) 998(1)Table 2 Comparison of extra13ted masses at g = 05



m

non-standard Wilson fermionstandard Wilson bosonstandard Wilson fermionSLAC bosonSLAC fermionnon-standard Wilson bosonnon-standard Wilson fermion

002 004 006 008 01 012 01466577588599510105

Figure 14 Ee13tive mass for the two dimensional Wess-Zumino model at g = 05

35

degrees of freedom the resulting Dira13 operators have a rather simple form a fa13t whi13h13an be put to pra13ti13al use as will be shown belowNonetheless the mere presen13e of the fermion determinant in the partition fun13tion intro-du13es a nonlo13ality whi13h has to be taken into a1313ount in the Monte Carlo algorithm Wethus de13ided to base our numeri13al simulations on the HMC whi13h allows for a simultane-ous update of all bosoni13 eld variables For the quantum me13hani13al models dis13ussed inse13tion 2 substantial improvements 13an be a13hieved if it is possible to 13ompute the fermiondeterminant in 13losed form For the two-dimensional models however 13omparable resultsare not available and these theories are plagued by a strongly u13tuating fermion determi-nant Even worse the fermion determinant may take on positive and negative values whi13hdrives the simulations dire13tly into the so-13alled sign problem To pro13eed we will thereforetreat the quantum me13hani13al models again separately from the two dimensional modelsand dis13uss them one at a time51 Quantum Me13hani13sOur setup for the bosoni13 degrees of freedom for the HMC does not dier from the standardpro13edure and is formulated on an enlarged phase spa13e involving the real bosoni13 eld φxand an additional 13onjugate momentum eld πx As usual these elds are propagated alongthe mole13ular dynami13s traje13tory by Hamiltons equationsφx =

partH

partπx πx = minus partH

partφx (87)where

H =1

2

sum

x

π2x + S[φ] (88)Sin13e on the latti13e the fermions are already integrated out the expression to be used ineq (88) is given by

S[φ] = SB[φ] + ln detM [φ] (89)In the standard approa13h one would now introdu13e a pseudofermion eld χ to obtain asto13hasti13 estimate for detM [φ] whi13h however will ne13essarily introdu13e additional noiseto later measurements Hen13e it would be 13learly favorable to take the fermion determinantexa13tly into a1313ount While a dire13t 13omputation of the fermion determinant at ea13h stepof a traje13tory is also feasible in these one-dimensional theories one 13an do even better dueto the simple stru13ture of the Dira13 operator Despite their dieren13es in the bosoni13 part36

of the a13tion the models involving an antisymmetri13 derivative matrix and a Wilson termnamely the models (i) (ii) and (iv) share the same fermion MatrixMW [φ] For these 13asesa 13losed expression for the determinant with Wilson parameter r = 1 is given by (29)detMW [φ] =

prod

x

(

1 +m+ 3gφ2x

)

minus 1 (90)It should be noted that with this `diagonal nonlo13al form of the determinant even lo13alalgorithms 13ould have been used A similar expression for the Stratonovi13h pres13ription asgiven in (32) also allows for a qui13k 13omputation of the fermion determinant in model (v)For the SLAC derivative it is easier to again exploit the lo13al stru13ture of the intera13tionterms in the fermion matrix by varyingδ(tr lnM [φ]) = tr(δM [φ]Mminus1[φ]) (91)in order to determine the 13ontribution of ln detM [φ] = tr lnM [φ] to the equations of mo-tion (87) Sin13e we only 13onsider Yukawa 13ouplings the variation δM [φ] enters (91) as

partM [φ]xypartφz

= 3gφzδxzδyz (92)This holds true if the derivative of the superpotential appears in M only on the diagonalthis means that only a single matrix element of Mminus1[φ] has to be 13omputed for ea13h siteφz14 Obviously we 13annot avoid an inversion of M [φ] in order to update all sitesFinally let us briey mention some details of our simulation runs Sin13e the physi13al valueof m as well as the physi13al volume of the box L were kept xed at L = 10 middot mminus1 =

Na the latti13e spa13ing a was varied with the number of points N in the latti13e Thelatti13e sizes we have 13onsidered range from N = 15 to N = 243 13orresponding to a =

006 0004 We have 13he13ked that the measurements were insensitive to nite size ee13tsFor all latti13e sizes we generated between 250 000 and 400 000 independent 13ongurationsfor the mass extra13tion in order to improve the signal to noise ratio for the Ward identities106 independent 13ongurations were used (with 4 independent runs in order to minimizestatisti13al errors)14If this method was to be applied to the Stratonovi13h pres13ription (where we 13an alternatively use (32))two matrix elements would have to be 13omputed sin13e the intera13tion is o-diagonal by one unit in thefermion matrix

37

52 The two-dimensional Wess-Zumino modelFor the two-dimensional models the fermion determinant is not positive denite in 13ontrastto the situation in one dimension Indeed a numeri13al experiment shows that the determi-nant has large u13tuations at strong 13oupling and 13hanges its sign if gm is larger than a13ertain threshold value of order O(1) This value depends on the latti13e spa13ing (and onm) A 13loser look at the spe13trum of the fermion matrix reveals a se13ond related problemnamely the existen13e of very small eigenvalues They in13rease the 13ondition number of thefermion matrix by several orders of magnitude and prevent a straightforward appli13ationof the pseudofermion method In addition the bosoni13 potential |W prime(φ)|2 possesses twoseparate minima this might lead to 13ompli13ations with respe13t to ergodi13ity In order to13he13k at whi13h values of the 13oupling the standard HMC breaks down we have performedsimulations in the weakly 13oupled regime where the aforementioned 13ontributions from thefermioni13 u13tuations 13an be taken into a1313ount via reweighting quen13hed ensembles Anadded advantage is that with this approa13h eld generations 13an be generated very qui13klyon the other hand larger ensembles are used sin13e reweighting requires higher statisti13s

g=2g=1

g=05g=02g=01

R

p(R

)

50-5-10-15-20-25

25

2

15

1

05

0Figure 15 Probability distribution of R = ln (detM detM0) The stronger pronoun13edthe peak the better the statisti13al errors are under 13ontrol The failure of the reweightingte13hnique is visible for g ge 1 The plotted data was generated from 20 000 13ongurationsfor the model (iii) on a 31 times 31 square latti13e38

quencheddynamical

t

Gbos(t

)

35302520151050

0045

004

0035

003

0025

002

0015

001

0005

0Figure 16 Comparison of the bosoni13 two-point fun13tion Gbos(t) for the quen13hed and fulltheory at g = 05 for the model with standard Wilson term (i) on a 32times 16 latti13e Similarresults hold for the other models as wellTrial runs indi13ate that for the volumes 13onsidered the reweighting method should work forg le 1 Figure 15 shows that for these moderate 13ouplings the u13tuations of the fermiondeterminant are within two orders of magnitude For g = 2 the u13tuations span more thanten orders of magnitude and the number of relevant 13ongurations be13omes ridi13ulouslysmallAs pointed out in se13tion 432 for g = 05 the perturbation of the mass is about 03 anee13t whi13h is 13learly not visible in our simulations However we 13an 13ompare quen13hedwith reweighted expe13tation values and 13he13k whether they are sensitive to the in13lusionof dynami13al fermions at all We have found that even in this weakly 13oupled regimethe ee13tive masses of bosoni13 and fermioni13 superpartners 13oin13ide only if the fermiondeterminant is properly taken into a1313ount Otherwise the two-point fun13tions (and hen13ethe masses) deviate 13onsiderably from their values in the full theory as 13an bee seen ingure 16 Clearly in order to simulate at larger 13ouplings an improved algorithm is neededFor example a 13omparison with the results in [30 would require a ratio of gm ≃ 03 whi13hobviously is not feasible with the reweighting method used in this paper

39

53 Measurements and determination of massesIn this subse13tion we expand on the extra13tion of ee13tive masses in our quantum me13hani13almodels They are determined by the asymptoti13 behavior of the (bosoni13 or fermioni13) two-point fun13tions G(x) ≃ sum

i cieminus(EiminusE0)x For x far away from the midpoint N2 of thelatti13e the asymptoti13 expression for the bosoni13 two-point fun13tion is dominated by asingle mass state with E1 minus E0 = meff and as a symmetri13 fun13tion it is proportional to

cosh(meff(xminusN2)) the fermioni13 two-point fun13tion as a superposition of symmetri13 andantisymmetri13 parts (w r t the midpoint) [30 is for small x proportional to eminusmeffx Thusthe masses 13an be extra13ted by exponential ts from the 13orresponding simulation datameff equiv log

(

G(x)

G(x+ a)

)

G = G(n)bos or G

(n)ferm (93)The x-region for the t should be 13hosen in su13h a way that(i) the 13ontribution of higher energy states with Ei gt E1 in the asymptoti13 expression

G(x) =sum

i cieminus(EiminusE0)x is negligible(ii) the ee13t of the improvement terms whi13h violate ree13tion positivity (i e the ci arenot ne13essarily positive) and therefore damp the two-point fun13tion for small valuesof x below the 13ontinuum value better not inuen13e the result(iii) the errors (whi13h are larger for smaller values of the two-point fun13tion) should beminimized(iv) and in the 13ase of bosons the inuen13e from the se13ond exponential tail in cosh(meff(xminus

N2)) = 12(emeff (xminusN2) + eminusmeff (xminusN2)) should not interfere with the exponential t tothe rstThus we are supposed to 13hoose a region for x in between the left boundary (x ≫ 0 by(i) and (ii)) and the midpoint (x ≪ N2 by (iii) and (iv)) This works reasonably wellfor Wilson fermions (the results are presented in se13tion 31) but for SLAC fermions weobserve an os13illating behavior of the two-point fun13tions near the boundaries whi13h requiresa more 13areful investigationIn order to understand the nature of this phenomenon we 13ompute the propagator of a freemassive fermion in the 13ontinuum

Gferm(x) =sum

λ

ψλ(x)ψlowastλ(0)

λ (94)40

where the sum runs over the nonzero part of the spe13trum of the dierential operator part+mand ψλ are the eigenfun13tions of part +m with eigenvalue λ If we impose periodi13 boundary13onditions with ψλ(x) = ψλ(x + N) we have ψλ(x) = 1radicNe

2πiNkx with λ = 2πi

Nk + m for allinteger k Inserting this into (94) we obtain

Gferm(x) =1

N

sum

k

e2πiNkx

2πiNk +m

(95)for the two-point fun13tion on the 13ir13le If we dis13retize the 13ir13le (and so introdu13e amomentum 13uto) the sum in (95) is trun13ated to a nite number of terms and redu13esjust to the propagator of the SLAC derivative This 13uto in momentum spa13e leads to theGibbs phenomenon whi13h is in fa13t what we observe e g in gure 17PSfrag repla13ements

latti13e point x

〈ψxψ

0〉

ltered fermioni13 twopointfun13tion fermioni13 twopointfun13tion (N = 61 m = 10 g = 0)ltered fermioni13 twopointfun13tion

0 10 20 30 40 50 60-010010203040506070809

Figure 17 The free fermioni13 two-point fun13tion (gphys = 0 mphys = 10) before and afterthe appli13ation of the lterThe error made by an approximation to a 13ontinuous and periodi13 fun13tion by n of itsFourier modes de13reases exponentially with n For a dis13ontinuous fun13tion the error isproportional to some power nminusδ away from the dis13ontinuities This 13an be improved by alter whi13h in13reases the 13onvergen13e rate δ or even re13overs the exponential approximation41

1e-09

1e-08

1e-07

1e-06

1e-05

1e-04

0001

001

01

1

0 20 40 60 80 100 120


latti13e point x

|〈ψxψ

0〉|

fermioni13 twopointfun13tion (N = 121 m = 10 g = 100)ltered fermioni13 twopointfun13tion

0001001011Figure 18 This plot shows one example of a fermioni13 two-point fun13tion of the intera13tingtheory (mphys = 10 gphys = 100) in a logarithmi13 representation before and after theappli13ation of the lter A linear t of the ltered fun13tion yields the ee13tive mass m Inthe last part of the graph a large numeri13al error 13an be observedof a 13ontinuous fun13tion15 For our 13omputations we have used the optimal lter proposedin [45 the lter has 13ompa13t support in a region away from the dis13ontinuities and is optimalin the sense that it balan13es the 13ompeting errors 13aused by lo13alizing it either in physi13alor in momentum spa13e Its ee13t in the intera13ting 13ase is illustrated in gure 18 In thislogarithmi13 plot it extends the range from whi13h one 13an extra13t the masses nearly up tothe midpoint of the latti13e The rather large deviations from the unltered data beyondthat point are irrelevant for our purposesFor the two-dimensional models we have extra13ted the masses from the two point 13orrela-tors [30Gbos(t) =

1

NtN2x

sum

tprime

sum

xxprime

〈ϕ2(t+ tprime x)ϕ2(tprime xprime)〉 (96)15In higher dimensions these Fourier approximation errors away from the dis13ontinuities are negligible in13omparison to other errors

42

andGferm(t) =

1

NtN2x

2sum

α=1

sum

tprime

sum

xxprime

〈ψα(t+ tprime x)ψα(tprime xprime)〉 (97)with Nt equiv N1 Nx equiv N2 Both 13orrelation fun13tions are proportional to cosh(m(tminusNt2))The mass 13an be extra13ted via a linear t in the logarithmi13 representation in analogy tothe one-dimensional 13ase6 Renormalizibility of the WZmodel with SLAC fermionsKartens and Smit have demonstrated in [46 that the SLAC derivative indu13es a nonrenor-malizable one-loop diagram in four-dimensional quantum ele13trodynami13s on the latti13eTherefore theories involving the SLAC derivative were generally believed to be nonrenor-malizable But it 13an be shown that the two-dimensional N = 2 Wess-Zumino model is infa13t renormalizable at least to one-loop order16As usual the 13al13ulations of latti13e perturbation theory are 13arried out in the thermodynami13limit where the number of latti13e points tends to innity and the latti13e momentum be13omes13ontinuous In momentum spa13e the nite latti13e spa13ing a is translated into a nite 13uto

Λ = πa It has to be shown that the diagrams 13an be renormalized when this 13uto isremoved The BPHZ renormalization s13heme is used to prove that the renormalized integralstend towards their 13ontinuum 13ounterparts and the 13ounterterms 13an be identied withsimilar quantities of 13ontinuum perturbation theory The argumentation employed here is13losely related to the renormalization theorem of Reisz [47 This theorem does howevernot apply in its original form be13ause the integrands are not smooth fun13tions of the loopmomentumIn the following we will determine an upper bound for the boson propagator in momentumspa13e whi13h will be used later on to argue that parts of the integrals in latti13e perturbationtheory are going to vanish in the 13ontinuum limit For the SLAC derivative the momentumspa13e representation of the propagators

1

P (k)2 +m2and minusi P (k) +m

P (k)2 +m2(98)for bosons and fermions 13ontains the saw tooth fun13tion

Pmicro(k) = kmicro minus 2lΛ where (2l minus 1)Λ le kmicro le (2l + 1)Λ (99)16In fa13t the dis13ussion 13an easily be extended to prove renormalizability also for the N = 1 model43

The momentum integration is always restri13ted to the rst Brillouin zone BZ = (kmicro)| |kmicro| leΛ The internal lines in the one-loop diagrams 13arry either the internal momentum k ora sum k + q of internal and external momenta where q denotes a linear 13ombination of theexternal momenta (using momentum 13onservation there are nminus1 su13h linear 13ombinationsqj in a diagram with n verti13es) Integrations over loop momenta kmicro 13an be split intointegrations over a square D = (kmicro)| |kmicro| le πε

a for an arbitrary ε lt 1

2and the rest ofthe Brillouin zone BZD We will argue below that the integral over D 13onverges to the13ontinuum value of the integral whereas the integral over BZD is shown to vanish as a goesto zeroNamely for a given set of external momenta qj one may 13hoose η = maxmicroja0π |qjmicro| with

a0 small enough su13h that 0 lt η lt ε lt 12 For (kmicro) isin D we 13an then read o from

a|kmicro plusmn qmicro| le a(|kmicro| + |qmicro|) lt π(ε+ η) for all a lt a0 (100)that |kmicroplusmn qmicro| le Λεprime with εprime = ε+ η lt 1 i e (kmicroplusmn qmicro) is also inside the rst Brillouin zoneOn the other hand if (kmicro) isin BZDπ(εminus η) le a(|kmicro| minus |qmicro|) le a|kmicro + qmicro| le a(|kmicro| + |qmicro|) le π(1 + η) (101)for su13h latti13e spa13ings a The latter inequality may be used in order to nd an upperbound for the propagator

1

P (k plusmn q)2 +m2lt

1

P (k plusmn q)2lt Ca2 (102)with C = ((εminus η)

radic2π)minus2It 13an be easily seen that in the Wess-Zumino model only two dierent types of integrals13ontribute at one-loop level A typi13al integral of the rst type is

Iε + Iπ =

int

BZ

d2k

(2π)2

1

(P (k)2 +m2)(P (k + q1)2 +m2) (P (k + qnminus1)2 +m2) (103)

Iε =

int

D

d2k

(2π)2

1

(k2 +m2)((k + q1)2 +m2) ((k + qnminus1)2 +m2)

Iπ =

int

BZD

d2k

(2π)2

1

(k2 +m2)(P (k + q1)2 +m2) (P (k + qnminus1)2 +m2)

le (Ca2)nminus1

int

|k|leradic

2πa

d2k

(2π)2

1

(k2 +m2)= (Ca2)nminus1 log

(

1 +2π2

a2m2

)

Here we have applied (101) in order to nd an upper bound for the integrand in Iπ andthen enlarged the integration domain to a full disk in13luding the rst Brillouin zone Thus44

Iπ vanishes in the 13ontinuum limit if n gt 1 Therefore the integral Iε tends to the 13ontin-uum value of the integral as a goes to zero (and the 13orresponding 13ontinuum integral is13onvergent by power 13ounting) so as long as we are 13onsidering diagrams with more thanone vertex this type of integrals does not spoil renormalizabilityAnother 13lass of integrals isI primeε + I primeπ =

int

BZ

d2k

(2π)2

Pmicro(k)Pν(k + q1) P(k + ql)


I primeε =

int

D

d2k

(2π)2

kmicro (k + ql)(k2 +m2)((k + q1)2 +m2) ((k + qnminus1)2 +m2)

I primeπ =

int

BZD

d2k

(2π)2

Pmicro(k) P(k + ql)


leint

BZD

d2k

(2π)2

|Pmicro(k)| |P(k + ql)|(k2 +m2)(P (k + q1)2 +m2) (P (k + qnminus1)2 +m2)

le(π

a

)l+1

(Ca2)nminus1

int

|k|leradic

2πa

d2k

(2π)2

1

(k2 +m2)= Cnminus1a2nminuslminus3 log

(

1 +2π2

a2m2

)

The qi are taken from the qj so l le n minus 1 The same arguments as above show that the13ontinuum limit is 13orre13t for any n gt 2 (again all 13ontinuum integrals are 13onvergent bypower 13ounting)Therefore renormalizability only remains to be shown for two kinds of integrals The rst13onsists of diagrams with n = 1 e g tadpole diagrams In this 13ase the loop momentumis independent of the (vanishing) exterior momentum so that the argument of Pmicro(k) isrestri13ted to the rst Brillouin zone (where Pmicro(k) = kmicro) The boundary of the integrationregion behaves as a nite 13uto that is removed in the 13ontinuum limit so that the integralapproa13hes its 13ontinuum 13ounterpart In the BPHZ renormalization s13heme these diagramsare just subtra13ted and do not 13ontribute to the renormalized quantitiesThe se13ond kind of integrals (with n = 2 and l = 1 in (104)) requires a more 13areful investi-gation whi13h may be found in appendix A This demontrates that latti13e dis13retizations ofthe two-dimensional N = 2 Wess-Zumino model based on the SLAC derivative are renor-malizable at rst order in perturbation theory and yield the 13orre13t 13ontinuum limit Itseems however problemati13 to use the BPHZ renormalization s13heme to renormalize the13orresponding diagrams in higher-dimensional 13ases sin13e this would require a dierentia-tion of the integrands with respe13t to external momenta The dis13ontinuity of the saw toothfun13tions in this 13ase would lead to singular terms45

7 Summary and outlookIn this paper we have studied supersymmetri13 N = 2 Wess-Zumino models in one andtwo dimensions The six quantum me13hani13al models under 13onsideration dier by the13hoi13e of latti13e derivatives and improvement terms The latter 13an be used to render thetheory manifestly supersymmetri13 on the latti13e in distin13tion to previous works on thissubje13t our simulations of the broken Ward identities at strong 13oupling prove that onlyone supersymmetry 13an be preserved We have demonstrated to a high numeri13al pre13isionthat by far the best results for bosoni13 and fermioni13 masses 13an be obtained from a modelwith Wilson fermions and Stratonovi13h pres13ription for the evaluaton of the improvementterm and from a model based on the SLAC derivative It is interesting to note that forSLAC fermions no improvement term is needed to re13over supersymmetry in the 13ontinuumlimitAs a key result of this paper for two-dimensional Wess-Zumino models we propose a non-standard Wilson term giving rise to an O(a4) improved Dira13 operatorM = γmicropartmicro +

iar

2γ3∆ with r2 =

4

3 (105)The masses extra13ted from this model approa13h the 13ontinuum values mu13h faster than thosefor the model with standard Wilson-Dira13 operator (61) Again results of a 13omparablygood quality 13an be obtained with nonlo13al SLAC fermions In our 13ase the 13ommonreservation that the SLAC derivative leads to non-renormalizable theories (as originallyshown in [46 for the 13ase of four-dimensional gauge theory) does not hold we have proventhat the Wess-Zumino model in two dimensions with this derivative is renormalizable toone-loop orderMotivated by the fa13t that the masses for the N = 2 Wess-Zumino latti13e model withultralo13al Dira13 operator (105) are quite 13lose to the 13ontinuum values already for moderatelatti13es we plan to study the model at strong 13ouplings where we will see deviations fromthe free theory We are about to implement the PHMC algorithm [48 as a possibility todeal with the small eigenvalues of the fermioni13 operator We believe that the N = 2Wess-Zumino model as a simple and well-understood theory without the 13ompli13ations ofgauge elds has the potential to be13ome a toy-model for developing e13ient algorithms forsystems with dynami13al fermions similar to the ubiquitous S13hwinger model whi13h servesas toy model for more 13omplex systems with a 13hiral 13ondensate instantons 13onnementand so forth 46

Wess-Zumino models are the at-spa13e limits of Landau-Ginzburg models A related proje13tmight be to study another limit of Landau-Ginzburg theories namely sigma models in non-trivial Kaumlhlerian without a superpotential Su13h sigma-models admit two supersymmetriestypi13ally they have instanton solutions and 13hiral 13ondensates may be generated If thereexist lo13al Ni13olai variables whi13h give rise to improved latti13e models with one quarter ofsupersymmetry 13onta13t 13ould be made with our investigations of Wess-Zumino models ina mu13h broader physi13al 13ontext Clearly these interesting eld theories deserve furtherattention both from the algebrai13 and from the numeri13al sideA further obvious problem is to study the nonperturbative se13tor of the two-dimensionalN = 1 Wess-Zumino model This model shows a ri13her phase stru13ture than the modelwith two supersymmetries The sign problem for the Pfaan seems unavoidable It isinteresting to note that in 13onventions with hermitean gamma matri13es a nonvanishingWilson term for Majorana fermions has to enter the Dira13 operator as in (105) Neverthelessthe dis13retization errors in this 13ase will be of order O(a) this 13an in prin13iple be improvedto O(a4) by a slightly less natural substitute for the Wilson term In general for the N = 1model no lo13al Ni13olai variables 13an be 13onstru13ted whi13h would suggest a supersymmetri1313ompletion of the naive latti13e a13tion However one might expe13t (as in quantum me13hani13s)that su13h improvement terms for the two-dimensional N = 1 model with SLAC fermionsare in fa13t dispensableA13knowledgementsWe thank F Bru13kmann S Catterall S Duerr C-P Georg J Giedt K Jansen andC Wozar for dis13ussions T Kaestner a13knowledges support by the Konrad-Adenauer-Stiftung and G Bergner by the Evangelis13hes Studienwerk This work has been supportedby the DFG grant Wi 7778-2A Renormalization of the fermion loopIn se13tion 6 we have demonstrated whi13h kinds of integrals are potentially dangerous forthe one-loop renormalizability of the N = 2 Wess-Zumino model The missing integral inthis proof was given by (104) with n = 2 and l = 1 it appears for fermion loops with two

47

internal linesq

k

propint

BZ

d2k

(2π)2

Pmicro(k)Pmicro(k minus q)

(P 2(k) +m2)(P 2(k minus q) +m2) (106)Depending on the superpotential more than two external lines may appear in this 13ase qdenotes the sum of all in13oming external momenta In order to fa13ilitate the evaluation ofthis integral we add (and subtra13t) a term with a nite 13ontinuum limit apart from thatthe integral needs to be regulated in this limit On the latti13e with nite latti13e spa13ingBPHZ regularization means to subtra13t the (as yet nite) value of the integral with vanishingexterior momenta Thus we 13onsider

int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)+

int

BZ

d2k

(2π)2

m2

(P 2(k) +m2)(P 2(k minus q) +m2)

minus (value at q = 0)

=

int

BZ

d2k

(2π)2

P micro(k minus q)(kmicro minus Pmicro(k minus q))

(P 2(k) +m2)(P 2(k minus q) +m2)

=

int

BZ

d2k

(2π)2

qmicroPmicro(k minus q)

(k2 +m2)(P (k minus q)2 +m2)

minus 2Λsum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2

P micro(k minus q)

(k2 +m2)(P (k minus q)2 +m2) (107)In the last step we have 13hosen a0 in su13h a way that for all a = πΛ lt a0 shifting kmicro isin BZby minusqmicro one winds up either in the same or in an adja13ent Brillouin zone i e

Pmicro(k minus q) = kmicro minus qmicro + 2Λ(Θ(minusΛ minus kmicro + qmicro) minus Θ(kmicro minus qmicro minus Λ)) (108)The rst term on the right-hand side of (107) 13an be easily seen to 13onverge to the valueof its 13ontinuum 13ounterpart by similar arguments as in (103) and (104) In order to provethat the se13ond term does not give rise to any 13orre13tions in the 13ontinuum limit we makeuse of (100) and (102) and observe that an upper bound for its modulus is given by2Λ

sum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2

|P micro(k minus q)|(k2 +m2)(P (k minus q)2 +m2)

le 2Cπ2

int minusΛ+q1

minusΛ

dk1

2π

int Λ

minusΛ

dk2

2π

1

k2 +m2+ (q1 harr q2 k1 harr k2)

=C

2

∣

∣

∣

∣

int Λ

minusΛ

dk2 arctan( q1ω(k2) minus q1Λω(k2)minus1 + Λ2ω(k2)minus1

)

ω(k2)minus1

∣

∣

∣

∣

+ (q1 harr q2 k1 harr k2)48

le C

2

int Λ

minusΛ

dk2

∣

∣

∣

q1m2 + k2

2 minus Λq1 + Λ2

∣

∣

∣+ (q1 harr q2 k1 harr k2) (109)with ω(k) =

radicm2 + k2 here we have also used that | arctan(x)| le |x| It is obvious thatthis upper bound 13onverges to zero in the limit where the latti13e 13uto is removedThis 13ompletes the proof that the dis13retization of the N = 2 Wess-Zumino model basedon the SLAC derivative is one-loop renormalizableReferen13es[1 E Witten Dynami13al breaking of supersymmetry Nu13l Phys B188 (1981) 513[2 P H Dondi and H Ni13olai Latti13e supersymmetry Nuovo Cim A41 (1977) 1[3 J Bartels and G Kramer A latti13e version of the Wess-Zumino model Z Phys C20(1983) 159[4 H B Nielsen and M Ninomiya Absen13e of neutrinos on a latti13e 1 Proof by homotopytheory Nu13l Phys B185 (1981) 20[5 H B Nielsen and M Ninomiya Absen13e of neutrinos on a latti13e 2 Intuitive topo-logi13al proof Nu13l Phys B193 (1981) 173[6 J Bartels and J B Bronzan Supersymmetry on a latti13e Phys Rev D28 (1983) 818[7 S Nojiri Continuous translation and supersymmetry on the latti13e Prog TheorPhys 74 (1985) 819[8 S Nojiri The spontaneous breakdown of supersymmetry on the nite latti13e ProgTheor Phys 74 (1985) 1124[9 A Kir13hberg J D Lange and A Wipf From the Dira13 operator to Wess-Zuminomodels on spatial latti13es Ann Phys 316 (2005) 357 [arXivhep-th0407207[10 M F L Golterman and D N Pet13her A lo13al intera13tive latti13e model with super-symmetry Nu13l Phys B319 (1989) 307[11 T Banks and P Windey Supersymmetri13 latti13e theories Nu13l Phys B198 (1982)226[12 K Fujikawa and M Ishibashi Latti13e 13hiral symmetry and the Wess-Zumino modelNu13l Phys B622 (2002) 115 [arXivhep-th0109156[13 K Fujikawa and M Ishibashi Latti13e 13hiral symmetry Yukawa 13ouplings and theMajorana 13ondition Phys Lett B528 (2002) 295 [arXivhep-lat011205049

[14 K Fujikawa Supersymmetry on the latti13e and the Leibniz rule Nu13l Phys B636(2002) 80 [arXivhep-th0205095[15 M Bonini and A Feo Wess-Zumino model with exa13t supersymmetry on the latti13eJHEP 09 (2004) 011 [arXivhep-lat0402034[16 M Bonini and A Feo Exa13t latti13e Ward-Takahashi identity for the N = 1 Wess-Zumino model Phys Rev D71 (2005) 114512 [arXivhep-lat0504010[17 S Catterall and S Karamov A latti13e study of the two-dimensional Wess-Zuminomodel Phys Rev D68 (2003) 014503 [arXivhep-lat0305002[18 S Elitzur E Rabinovi13i and A S13hwimmer Supersymmetri13 models on the latti13ePhys Lett B119 (1982) 165[19 V Rittenberg and S Yankielowi13z Supersymmetry on the latti13e CERN-TH-3263[20 N Sakai and M Sakamoto Latti13e supersymmetry and the Ni13olai mapping Nu13lPhys B229 (1983) 173[21 J Ranft and A S13hiller Hamiltonian Monte Carlo study of (1+1)-dimensional modelswith restri13ted supersymmetry on the latti13e Phys Lett B138 (1984) 166[22 A S13hiller and J Ranft The (1+1)-dimensional N = 2 Wess-Zumino model on thelatti13e in the lo13al hamiltonian method J Phys G12 (1986) 935[23 M Be1313aria and C Rampino World-line path integral study of supersym-metry breaking in the Wess-Zumino model Phys Rev D67 (2003) 127701[arXivhep-lat0303021[24 M Be1313aria M Campostrini and A Feo Supersymmetry breaking in two dimen-sions The latti13e N = 1 Wess-Zumino model Phys Rev D69 (2004) 095010[arXivhep-lat0402007[25 W Bietenholz Exa13t supersymmetry on the latti13e Mod Phys Lett A14 (1999) 51[arXivhep-lat9807010[26 H Aratyn P F Bessa and A H Zimerman On the two-dimensional N = 2 Wess-Zumino model on the latti13e From Dira13-Kahler formalism to the Ni13olai mapping ZPhys C27 (1985) 535[27 H Ni13olai On a new 13hara13terization of s13alar supersymmetri13 theories Phys LettB89 (1980) 341[28 S Ce13otti and L Girardello Lo13al Ni13olai mappings in extended supersymmetryIC82105 50

[29 S Elitzur and A S13hwimmer N = 2 two-dimensional Wess-Zumino model on thelatti13e Nu13l Phys B226 (1983) 109[30 S Catterall and S Karamov Exa13t latti13e supersymmetry The two-dimensional N =

2 Wess-Zumino model Phys Rev D65 (2002) 094501 [arXivhep-lat0108024[31 J Giedt R-symmetry in the Q-exa13t (22) 2d latti13e Wess-Zumino model Nu13l PhysB726 (2005) 210 [arXivhep-lat0507016[32 K Fujikawa N = 2 Wess-Zumino model on the d = 2 Eu13lidean latti13e Phys RevD66 (2002) 074510 [arXivhep-lat0208015[33 D Birmingham M Blau M Rakowski and G Thompson Topologi13al eld theoryPhys Rept 209 (1991) 129[34 S Catterall Latti13e supersymmetry and topologi13al eld theory JHEP 05 (2003) 038[arXivhep-lat0301028[35 J Giedt and E Poppitz Latti13e supersymmetry superelds and renormalization JHEP09 (2004) 029 [arXivhep-th0407135[36 S Catterall and E Gregory A latti13e path integral for supersymmetri13 quantum me-13hani13s Phys Lett B487 (2000) 349 [arXivhep-lat0006013[37 J Giedt R Koniuk E Poppitz and T Yavin Less naive about supersymmetri13 latti13equantum me13hani13s JHEP 12 (2004) 033 [arXivhep-lat0410041[38 H Ezawa and J R Klauder Fermion without fermions The Ni13olai map revisitedProg Theor Phys 74 (1985) 904[39 S D Drell M Weinstein and S Yankielowi13z Variational approa13h to strong 13ouplingeld theory 1 φ4 theory Phys Rev D14 (1976) 487[40 M Be1313aria G Cur13i and E DAmbrosio Simulation of supersymmetri13 models witha lo13al Ni13olai map Phys Rev D58 (1998) 065009 [arXivhep-lat9804010[41 I Montvay and G Muumlnster Quantum elds on a latti13e Cambridge Univ Pr (1994)491 p (Cambridge monographs on mathemati13al physi13s)[42 S Ce13otti and L Girardello Fun13tional measure topology and dynami13al supersym-metry breaking Phys Lett B110 (1982) 39[43 E Gozzi Fun13tional integral approa13h to Parisi-Wu sto13hasti13 quantization S13alartheory Phys Rev D28 (1983) 1922[44 S Duane A D Kennedy B J Pendleton and D Roweth Hybrid Monte Carlo PhysLett B195 (1987) 216 51

[45 J Tanner Optimal lter and mollier for pie13ewise smooth spe13tral data Mathemati13sof Computation 75 (2006) 767[46 L H Karsten and J Smit The va13uum polarization with SLAC latti13e fermions PhysLett B85 (1979) 100[47 T Reisz A power 13ounting theorem for Feynman integrals on the latti13e CommunMath Phys 116 (1988) 81[48 R Frezzotti and K Jansen A polynomial hybrid Monte Carlo algorithm Phys LettB402 (1997) 328 [arXivhep-lat9702016

52

Introduction

Supersymmetric quantum mechanics

Lattice models

Lattice models without improvement

The unimproved models in detail

Lattice models with improvement

The improved models in detail

Fermion determinants

Sign of the determinants

Calculating the determinants

Simulations of supersymmetric quantum mechanics

Effective Masses on the lattice

Models without interaction

Models with interaction

Ward identities

Ward identities of the unimproved models

Ward identities of the improved models

The N=2 Wess-Zumino model in 2 dimensions


The lattice models in detail

Simulations of the Wess-Zumino model



Algorithmic aspects

Quantum Mechanics

The two-dimensional Wess-Zumino model

Measurements and determination of masses

Renormalizibility of the WZ model with SLAC fermions

Summary and outlook

Renormalization of the fermion loop

1 Introdu13tionSupersymmetry is an important ingredient of modern high energy physi13s beyond the stan-dard model sin13e boson masses are prote13ted by supersymmetry in su13h theories with 13hiralfermions it helps to redu13e the hierar13hy and ne-tuning problems drasti13ally and withingrand unied theories it leads to predi13tions of the proton life-time in agreement withpresent day experimental bounds As low energy physi13s is manifestly not supersymmet-ri13 this symmetry has to be broken at some energy s13ale However non-renormalizationtheorems in four dimensions ensure that tree level supersymmetri13 theories preserve super-symmetry at any nite order of perturbation theory therefore supersymmetry has to bebroken non-perturbatively [1The latti13e formulation of quantum eld theories provides a systemati13 tool to investigatenon-perturbative problems In the 13ase of supersymmetri13 eld theories their formulation ishampered by the fa13t that the supersymmetry algebra 13loses on the generator of innitesimaltranslations [2 Sin13e Poin13areacute symmetry is expli13itly broken by the dis13retization oneis tempted to modify the supersymmetry algebra so as to 13lose on dis13rete translationsHowever as latti13e derivatives do not satisfy the Leibniz rule supersymmetri13 a13tions forintera13ting theories will in general not be invariant under su13h latti13e supersymmetries Theviolation of the Leibniz rule is an O(a) ee13t and supersymmetry naively will therefore berestored in the 13ontinuum limit In the 13ase of Poin13areacute symmetry the dis13rete remnants ofthe symmetry on the latti13e are su13ient to prohibit the appearan13e of relevant operators inthe ee13tive a13tion whi13h are invariant only under a subset of the Poin13areacute group and requirene tuning of their 13oe13ients in order to arrive at an invariant 13ontinuum limit In generi13latti13e formulations there are no dis13rete remnants of supersymmetry transformations onthe latti13e in su13h theories supersymmetry in the 13ontinuum limit 13an only be a13hieved byappropriately ne-tuning the bare 13ouplings of all supersymmetry-breaking 13ounterterms [3An additional 13ompli13ation in the formulation of supersymmetri13 theories on the latti13e isthe fermion doubling problem Lo13al and translationally invariant hermitean Dira13 opera-tors on the latti13e automati13ally des13ribe fermions of both 13hiralities [4 5 the fermioni13extra degrees of freedom are usually not paired with bosoni13 modes and so lead to supersym-metry breaking Generi13 pres13riptions eliminating these extra fermioni13 modes also breaksupersymmetryAs a simple supersymmetri13 theory the Wess-Zumino model in two dimensions has beenthe subje13t of intensive analyti13 and numeri13al investigations Early attempts in13luded1







int

dτ(1

2φ2 +

1


)


dφ (1)5



m2φ2 + g



1



(minus1)xminusy

a

πN



1




2φ2 + g



a

2∆xy =

1


Snaive =a

2

sum

x

(

(partφ)2x +W 2

x

)

+ asum

xy










a

2

sum

x

ξx(φ)2 + asum

xy

ψxpartξxpartφy


Ssusy =a

2

sum

x


sum

xy


= Snaive + asum

x



a

2

sum

x

ξx(φ)2 + asum

xy

ψxpartξypartφx


Ssusy =a

2

sum

x


sum

xy


= Snaive minus asum

x






sum






2∆xy +

partW prime(σx)


sum

x12

(




a






sum

xy


xy

vx(

part sxy +Wxy

)


sum

x


λ = minusar2

sum

xy

vx∆xyvy +sum

v



int

perbc

DψDψDφ eminusS =

int



(


partτ +m

)

=sinh (1

2

int β


sinh(β2m)



(

part + (Wxy)

partb +m1 )

Ito

=

prod




(

part + (Wxy)

partb +m1 )

Ito


e1

2βm2

det

(


partτ +m

) (30)sin13e lnprod




xy +1


det

(

part + (Wxy)

partb + (mxy)

)

Strat

=

prod

(1 + a2Mx) minus

prod

(1 minus a2Mx)

prod


(1 minus a2m)


(


partτ +m

)



(2)1d and S


1d S(5)1d and S




Sfree =1

2

sum

xy

φxKxyφy +sum

xy


(1)free S

(4)free = S

(2)free S

(5)free S

(6)free = S

(3)free




1 minus (am2

)2)


= MTM = MTM = MTM


4a2∆2 = minus∆14





m

m(1)bos

m(1)ferm

m(5)ferm

m(6)ferm

00102030405060708091

0 0005 001 0015 002 0025

001020304050607080918899294969810




m

naive SLAC fermions


00102030405060708091

0005 001 0015 002 0025

0010203040506070809112131415161718


(4)ferm(xm) =

(

1 minus am

2

)

G(5)ferm

(

xm(

1 minus am

2

))



m(124)ferm (a) = m

(24)bos (a) asymp m

(

1 minus am

2

)


W (φ) =m

2φ2 +

g





bos(0) =




(5)bos(0) = 1678 plusmn 004 and m




m



00102030405060708091

0005 001 0015 002 0025 003

001020304050607080911212513135141451515516165




m



00102030405060708091

0005 001 0015 002 0025 003

00102030405060708091121251313514145151551616517




m



00102030405060708091

0005 001 0015 002 0025 003

00102030405060708091121251313514145151551616517














m


improved SLAC ferm


00102030405060708091

0 0005 001 0015 002 0025 003 0035

00102030405060708091121251313514145151551616517


int

D(φ ψ) eminusS+P


sum

y


(41)20


latti13e point x

R(1

)x

R(2)

x


0 5 10 15 20-0002-0001000010002




latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708090001-0002-0001000010002



latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708090001-0006-0004-00020000200040006







latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708090001-0006-0004-00020000200040006


2)〉 = 0






latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708091-002-0015-001-0005000050010015002



latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708091-002-0015-001-0005000050010015002




latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708091-0025-002-0015-001-00050000500100150020025




int

d2x(

2partφpartφ +1

2|W prime|2 + ψMψ

)


ψ =

(

ψ1

ψ2






2W primeε2





1

2a

(






1

a2

(

sum

micro=12




2

sum


sum

x

(


1

2|Wx|2

)







)

=

(

Wxy 2partxy

2partxy Wxy

)

Wxy =partWx

partφy (57)28


sum

xy




2εξx δ(1)ψ1

x = 0









ar


S(1)bos =

a2

2

sum

x

(


+ a2sum

x

(




2(amr) a2

sum


(1)bos =

a2

2

sum

xy


pmicro =1


2

asin

(apmicro2





2arp2 plusmn i|





iar



iar


S(2)bos =

a2

2

sum

x

(


+ a2sum

x

(



S(2)bos =

a2

2

sum

xy


microp = m2 +

p2 +(

12ar p2


radic

p2 +(

12arp2




r2 =4

3(75)31





(3)bos =

a2

2

sum

xy


ϕ =

(

ϕ1

ϕ2

) and ξ =

(

ξ1

ξ2

)


2mφ2 + 1


W prime(φ) = mϕ1 + u+ i(mϕ2 + v) (80)32



(n)0 +m)ϕ +

(

u

v


S(n)bos =

a2

2

sum

xy

(ϕx K(n)xy ϕy) +

a2

2

sum

x


sum

x

ux

(

(div ϕ)x minusar

2(∆ϕ1)x

)

minus a2sum

x

vx

(

(13url ϕ)x +ar

2(∆ϕ2)x

)

∆(2) = a2sum

x

ux

(

(div ϕ)x minusar

2(∆ϕ2)x

)

+ a2sum

x

vx

(


2(∆ϕ1)x

)

(84)∆(3) = a2

sum

x


x



1

2mφ2 +

1


m1minusloop = m(

1 minus g2

4πm2






m


002 004 006 008 01 012 01466577588599510105


35


partH


partφx (87)where

H =1

2

sum

x




prod

x

(

1 +m+ 3gφ2x

)


partM [φ]xypartφz




37


g=2g=1

g=05g=02g=01

R

p(R

)

50-5-10-15-20-25

25

2

15

1

05


quencheddynamical

t

Gbos(t

)

35302520151050

0045

004

0035

003

0025

002

0015

001

0005


39




(

G(x)

G(x+ a)

)

G = G(n)bos or G


G(x) =sum



Gferm(x) =sum

λ


λ (94)40




Gferm(x) =1

N

sum

k

e2πiNkx

2πiNk +m


latti13e point x

〈ψxψ

0〉


0 10 20 30 40 50 60-010010203040506070809


1e-09

1e-08

1e-07

1e-06

1e-05

1e-04

0001

001

01

1

0 20 40 60 80 100 120


latti13e point x

|〈ψxψ

0〉|



1

NtN2x

sum

tprime

sum

xxprime


42

andGferm(t) =

1

NtN2x

2sum

α=1

sum

tprime

sum

xxprime



1









1


1



Iε + Iπ =

int

BZ

d2k

(2π)2

1


Iε =

int

D

d2k

(2π)2

1

(k2 +m2)((k + q1)2 +m2) ((k + qnminus1)2 +m2)

Iπ =

int

BZD

d2k

(2π)2

1


le (Ca2)nminus1

int

|k|leradic

2πa

d2k

(2π)2

1


(

1 +2π2

a2m2

)



int

BZ

d2k

(2π)2



I primeε =

int

D

d2k

(2π)2


I primeπ =

int

BZD

d2k

(2π)2

Pmicro(k) P(k + ql)


leint

BZD

d2k

(2π)2


le(π

a

)l+1

(Ca2)nminus1

int

|k|leradic

2πa

d2k

(2π)2

1


(

1 +2π2

a2m2

)



iar

2γ3∆ with r2 =

4



47

internal linesq

k

propint

BZ

d2k

(2π)2



int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)+

int

BZ

d2k

(2π)2

m2

(P 2(k) +m2)(P 2(k minus q) +m2)


=

int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)

=

int

BZ

d2k

(2π)2


(k2 +m2)(P (k minus q)2 +m2)

minus 2Λsum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2

P micro(k minus q)



sum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2


le 2Cπ2

int minusΛ+q1

minusΛ

dk1

2π

int Λ

minusΛ

dk2

2π

1


=C

2

∣

∣

∣

∣

int Λ

minusΛ


)

ω(k2)minus1

∣

∣

∣

∣


le C

2

int Λ

minusΛ

dk2

∣

∣

∣

q1m2 + k2

2 minus Λq1 + Λ2

∣

∣







52

Introduction


Lattice models












Ward identities









Algorithmic aspects

Quantum Mechanics




Summary and outlook








int

dτ(1

2φ2 +

1


)


dφ (1)5



m2φ2 + g



1



(minus1)xminusy

a

πN



1




2φ2 + g



a

2∆xy =

1


Snaive =a

2

sum

x

(

(partφ)2x +W 2

x

)

+ asum

xy










a

2

sum

x

ξx(φ)2 + asum

xy

ψxpartξxpartφy


Ssusy =a

2

sum

x


sum

xy


= Snaive + asum

x



a

2

sum

x

ξx(φ)2 + asum

xy

ψxpartξypartφx


Ssusy =a

2

sum

x


sum

xy


= Snaive minus asum

x






sum






2∆xy +

partW prime(σx)


sum

x12

(




a






sum

xy


xy

vx(

part sxy +Wxy

)


sum

x


λ = minusar2

sum

xy

vx∆xyvy +sum

v



int

perbc

DψDψDφ eminusS =

int



(


partτ +m

)

=sinh (1

2

int β


sinh(β2m)



(

part + (Wxy)

partb +m1 )

Ito

=

prod




(

part + (Wxy)

partb +m1 )

Ito


e1

2βm2

det

(


partτ +m

) (30)sin13e lnprod




xy +1


det

(

part + (Wxy)

partb + (mxy)

)

Strat

=

prod

(1 + a2Mx) minus

prod

(1 minus a2Mx)

prod


(1 minus a2m)


(


partτ +m

)



(2)1d and S


1d S(5)1d and S




Sfree =1

2

sum

xy

φxKxyφy +sum

xy


(1)free S

(4)free = S

(2)free S

(5)free S

(6)free = S

(3)free




1 minus (am2

)2)


= MTM = MTM = MTM


4a2∆2 = minus∆14





m

m(1)bos

m(1)ferm

m(5)ferm

m(6)ferm

00102030405060708091

0 0005 001 0015 002 0025

001020304050607080918899294969810




m

naive SLAC fermions


00102030405060708091

0005 001 0015 002 0025

0010203040506070809112131415161718


(4)ferm(xm) =

(

1 minus am

2

)

G(5)ferm

(

xm(

1 minus am

2

))



m(124)ferm (a) = m

(24)bos (a) asymp m

(

1 minus am

2

)


W (φ) =m

2φ2 +

g





bos(0) =




(5)bos(0) = 1678 plusmn 004 and m




m



00102030405060708091

0005 001 0015 002 0025 003

001020304050607080911212513135141451515516165




m



00102030405060708091

0005 001 0015 002 0025 003

00102030405060708091121251313514145151551616517




m



00102030405060708091

0005 001 0015 002 0025 003

00102030405060708091121251313514145151551616517














m


improved SLAC ferm


00102030405060708091

0 0005 001 0015 002 0025 003 0035

00102030405060708091121251313514145151551616517


int

D(φ ψ) eminusS+P


sum

y


(41)20


latti13e point x

R(1

)x

R(2)

x


0 5 10 15 20-0002-0001000010002




latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708090001-0002-0001000010002



latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708090001-0006-0004-00020000200040006







latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708090001-0006-0004-00020000200040006


2)〉 = 0






latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708091-002-0015-001-0005000050010015002



latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708091-002-0015-001-0005000050010015002




latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708091-0025-002-0015-001-00050000500100150020025




int

d2x(

2partφpartφ +1

2|W prime|2 + ψMψ

)


ψ =

(

ψ1

ψ2






2W primeε2





1

2a

(






1

a2

(

sum

micro=12




2

sum


sum

x

(


1

2|Wx|2

)







)

=

(

Wxy 2partxy

2partxy Wxy

)

Wxy =partWx

partφy (57)28


sum

xy




2εξx δ(1)ψ1

x = 0









ar


S(1)bos =

a2

2

sum

x

(


+ a2sum

x

(




2(amr) a2

sum


(1)bos =

a2

2

sum

xy


pmicro =1


2

asin

(apmicro2





2arp2 plusmn i|





iar



iar


S(2)bos =

a2

2

sum

x

(


+ a2sum

x

(



S(2)bos =

a2

2

sum

xy


microp = m2 +

p2 +(

12ar p2


radic

p2 +(

12arp2




r2 =4

3(75)31





(3)bos =

a2

2

sum

xy


ϕ =

(

ϕ1

ϕ2

) and ξ =

(

ξ1

ξ2

)


2mφ2 + 1


W prime(φ) = mϕ1 + u+ i(mϕ2 + v) (80)32



(n)0 +m)ϕ +

(

u

v


S(n)bos =

a2

2

sum

xy

(ϕx K(n)xy ϕy) +

a2

2

sum

x


sum

x

ux

(

(div ϕ)x minusar

2(∆ϕ1)x

)

minus a2sum

x

vx

(

(13url ϕ)x +ar

2(∆ϕ2)x

)

∆(2) = a2sum

x

ux

(

(div ϕ)x minusar

2(∆ϕ2)x

)

+ a2sum

x

vx

(


2(∆ϕ1)x

)

(84)∆(3) = a2

sum

x


x



1

2mφ2 +

1


m1minusloop = m(

1 minus g2

4πm2






m


002 004 006 008 01 012 01466577588599510105


35


partH


partφx (87)where

H =1

2

sum

x




prod

x

(

1 +m+ 3gφ2x

)


partM [φ]xypartφz




37


g=2g=1

g=05g=02g=01

R

p(R

)

50-5-10-15-20-25

25

2

15

1

05


quencheddynamical

t

Gbos(t

)

35302520151050

0045

004

0035

003

0025

002

0015

001

0005


39




(

G(x)

G(x+ a)

)

G = G(n)bos or G


G(x) =sum



Gferm(x) =sum

λ


λ (94)40




Gferm(x) =1

N

sum

k

e2πiNkx

2πiNk +m


latti13e point x

〈ψxψ

0〉


0 10 20 30 40 50 60-010010203040506070809


1e-09

1e-08

1e-07

1e-06

1e-05

1e-04

0001

001

01

1

0 20 40 60 80 100 120


latti13e point x

|〈ψxψ

0〉|



1

NtN2x

sum

tprime

sum

xxprime


42

andGferm(t) =

1

NtN2x

2sum

α=1

sum

tprime

sum

xxprime



1









1


1



Iε + Iπ =

int

BZ

d2k

(2π)2

1


Iε =

int

D

d2k

(2π)2

1

(k2 +m2)((k + q1)2 +m2) ((k + qnminus1)2 +m2)

Iπ =

int

BZD

d2k

(2π)2

1


le (Ca2)nminus1

int

|k|leradic

2πa

d2k

(2π)2

1


(

1 +2π2

a2m2

)



int

BZ

d2k

(2π)2



I primeε =

int

D

d2k

(2π)2


I primeπ =

int

BZD

d2k

(2π)2

Pmicro(k) P(k + ql)


leint

BZD

d2k

(2π)2


le(π

a

)l+1

(Ca2)nminus1

int

|k|leradic

2πa

d2k

(2π)2

1


(

1 +2π2

a2m2

)



iar

2γ3∆ with r2 =

4



47

internal linesq

k

propint

BZ

d2k

(2π)2



int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)+

int

BZ

d2k

(2π)2

m2

(P 2(k) +m2)(P 2(k minus q) +m2)


=

int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)

=

int

BZ

d2k

(2π)2


(k2 +m2)(P (k minus q)2 +m2)

minus 2Λsum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2

P micro(k minus q)



sum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2


le 2Cπ2

int minusΛ+q1

minusΛ

dk1

2π

int Λ

minusΛ

dk2

2π

1


=C

2

∣

∣

∣

∣

int Λ

minusΛ


)

ω(k2)minus1

∣

∣

∣

∣


le C

2

int Λ

minusΛ

dk2

∣

∣

∣

q1m2 + k2

2 minus Λq1 + Λ2

∣

∣







52

Introduction


Lattice models












Ward identities









Algorithmic aspects

Quantum Mechanics




Summary and outlook




m2φ2 + g



1



(minus1)xminusy

a

πN



1




2φ2 + g



a

2∆xy =

1


Snaive =a

2

sum

x

(

(partφ)2x +W 2

x

)

+ asum

xy










a

2

sum

x

ξx(φ)2 + asum

xy

ψxpartξxpartφy


Ssusy =a

2

sum

x


sum

xy


= Snaive + asum

x



a

2

sum

x

ξx(φ)2 + asum

xy

ψxpartξypartφx


Ssusy =a

2

sum

x


sum

xy


= Snaive minus asum

x






sum






2∆xy +

partW prime(σx)


sum

x12

(




a






sum

xy


xy

vx(

part sxy +Wxy

)


sum

x


λ = minusar2

sum

xy

vx∆xyvy +sum

v



int

perbc

DψDψDφ eminusS =

int



(


partτ +m

)

=sinh (1

2

int β


sinh(β2m)



(

part + (Wxy)

partb +m1 )

Ito

=

prod




(

part + (Wxy)

partb +m1 )

Ito


e1

2βm2

det

(


partτ +m

) (30)sin13e lnprod




xy +1


det

(

part + (Wxy)

partb + (mxy)

)

Strat

=

prod

(1 + a2Mx) minus

prod

(1 minus a2Mx)

prod


(1 minus a2m)


(


partτ +m

)



(2)1d and S


1d S(5)1d and S




Sfree =1

2

sum

xy

φxKxyφy +sum

xy


(1)free S

(4)free = S

(2)free S

(5)free S

(6)free = S

(3)free




1 minus (am2

)2)


= MTM = MTM = MTM


4a2∆2 = minus∆14





m

m(1)bos

m(1)ferm

m(5)ferm

m(6)ferm

00102030405060708091

0 0005 001 0015 002 0025

001020304050607080918899294969810




m

naive SLAC fermions


00102030405060708091

0005 001 0015 002 0025

0010203040506070809112131415161718


(4)ferm(xm) =

(

1 minus am

2

)

G(5)ferm

(

xm(

1 minus am

2

))



m(124)ferm (a) = m

(24)bos (a) asymp m

(

1 minus am

2

)


W (φ) =m

2φ2 +

g





bos(0) =




(5)bos(0) = 1678 plusmn 004 and m




m



00102030405060708091

0005 001 0015 002 0025 003

001020304050607080911212513135141451515516165




m



00102030405060708091

0005 001 0015 002 0025 003

00102030405060708091121251313514145151551616517




m



00102030405060708091

0005 001 0015 002 0025 003

00102030405060708091121251313514145151551616517














m


improved SLAC ferm


00102030405060708091

0 0005 001 0015 002 0025 003 0035

00102030405060708091121251313514145151551616517


int

D(φ ψ) eminusS+P


sum

y


(41)20


latti13e point x

R(1

)x

R(2)

x


0 5 10 15 20-0002-0001000010002




latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708090001-0002-0001000010002



latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708090001-0006-0004-00020000200040006







latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708090001-0006-0004-00020000200040006


2)〉 = 0






latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708091-002-0015-001-0005000050010015002



latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708091-002-0015-001-0005000050010015002




latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708091-0025-002-0015-001-00050000500100150020025




int

d2x(

2partφpartφ +1

2|W prime|2 + ψMψ

)


ψ =

(

ψ1

ψ2






2W primeε2





1

2a

(






1

a2

(

sum

micro=12




2

sum


sum

x

(


1

2|Wx|2

)







)

=

(

Wxy 2partxy

2partxy Wxy

)

Wxy =partWx

partφy (57)28


sum

xy




2εξx δ(1)ψ1

x = 0









ar


S(1)bos =

a2

2

sum

x

(


+ a2sum

x

(




2(amr) a2

sum


(1)bos =

a2

2

sum

xy


pmicro =1


2

asin

(apmicro2





2arp2 plusmn i|





iar



iar


S(2)bos =

a2

2

sum

x

(


+ a2sum

x

(



S(2)bos =

a2

2

sum

xy


microp = m2 +

p2 +(

12ar p2


radic

p2 +(

12arp2




r2 =4

3(75)31





(3)bos =

a2

2

sum

xy


ϕ =

(

ϕ1

ϕ2

) and ξ =

(

ξ1

ξ2

)


2mφ2 + 1


W prime(φ) = mϕ1 + u+ i(mϕ2 + v) (80)32



(n)0 +m)ϕ +

(

u

v


S(n)bos =

a2

2

sum

xy

(ϕx K(n)xy ϕy) +

a2

2

sum

x


sum

x

ux

(

(div ϕ)x minusar

2(∆ϕ1)x

)

minus a2sum

x

vx

(

(13url ϕ)x +ar

2(∆ϕ2)x

)

∆(2) = a2sum

x

ux

(

(div ϕ)x minusar

2(∆ϕ2)x

)

+ a2sum

x

vx

(


2(∆ϕ1)x

)

(84)∆(3) = a2

sum

x


x



1

2mφ2 +

1


m1minusloop = m(

1 minus g2

4πm2






m


002 004 006 008 01 012 01466577588599510105


35


partH


partφx (87)where

H =1

2

sum

x




prod

x

(

1 +m+ 3gφ2x

)


partM [φ]xypartφz




37


g=2g=1

g=05g=02g=01

R

p(R

)

50-5-10-15-20-25

25

2

15

1

05


quencheddynamical

t

Gbos(t

)

35302520151050

0045

004

0035

003

0025

002

0015

001

0005


39




(

G(x)

G(x+ a)

)

G = G(n)bos or G


G(x) =sum



Gferm(x) =sum

λ


λ (94)40




Gferm(x) =1

N

sum

k

e2πiNkx

2πiNk +m


latti13e point x

〈ψxψ

0〉


0 10 20 30 40 50 60-010010203040506070809


1e-09

1e-08

1e-07

1e-06

1e-05

1e-04

0001

001

01

1

0 20 40 60 80 100 120


latti13e point x

|〈ψxψ

0〉|



1

NtN2x

sum

tprime

sum

xxprime


42

andGferm(t) =

1

NtN2x

2sum

α=1

sum

tprime

sum

xxprime



1









1


1



Iε + Iπ =

int

BZ

d2k

(2π)2

1


Iε =

int

D

d2k

(2π)2

1

(k2 +m2)((k + q1)2 +m2) ((k + qnminus1)2 +m2)

Iπ =

int

BZD

d2k

(2π)2

1


le (Ca2)nminus1

int

|k|leradic

2πa

d2k

(2π)2

1


(

1 +2π2

a2m2

)



int

BZ

d2k

(2π)2



I primeε =

int

D

d2k

(2π)2


I primeπ =

int

BZD

d2k

(2π)2

Pmicro(k) P(k + ql)


leint

BZD

d2k

(2π)2


le(π

a

)l+1

(Ca2)nminus1

int

|k|leradic

2πa

d2k

(2π)2

1


(

1 +2π2

a2m2

)



iar

2γ3∆ with r2 =

4



47

internal linesq

k

propint

BZ

d2k

(2π)2



int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)+

int

BZ

d2k

(2π)2

m2

(P 2(k) +m2)(P 2(k minus q) +m2)


=

int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)

=

int

BZ

d2k

(2π)2


(k2 +m2)(P (k minus q)2 +m2)

minus 2Λsum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2

P micro(k minus q)



sum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2


le 2Cπ2

int minusΛ+q1

minusΛ

dk1

2π

int Λ

minusΛ

dk2

2π

1


=C

2

∣

∣

∣

∣

int Λ

minusΛ


)

ω(k2)minus1

∣

∣

∣

∣


le C

2

int Λ

minusΛ

dk2

∣

∣

∣

q1m2 + k2

2 minus Λq1 + Λ2

∣

∣







52

Introduction


Lattice models












Ward identities









Algorithmic aspects

Quantum Mechanics




Summary and outlook



2φ2 + g



a

2∆xy =

1


Snaive =a

2

sum

x

(

(partφ)2x +W 2

x

)

+ asum

xy










a

2

sum

x

ξx(φ)2 + asum

xy

ψxpartξxpartφy


Ssusy =a

2

sum

x


sum

xy


= Snaive + asum

x



a

2

sum

x

ξx(φ)2 + asum

xy

ψxpartξypartφx


Ssusy =a

2

sum

x


sum

xy


= Snaive minus asum

x






sum






2∆xy +

partW prime(σx)


sum

x12

(




a






sum

xy


xy

vx(

part sxy +Wxy

)


sum

x


λ = minusar2

sum

xy

vx∆xyvy +sum

v



int

perbc

DψDψDφ eminusS =

int



(


partτ +m

)

=sinh (1

2

int β


sinh(β2m)



(

part + (Wxy)

partb +m1 )

Ito

=

prod




(

part + (Wxy)

partb +m1 )

Ito


e1

2βm2

det

(


partτ +m

) (30)sin13e lnprod




xy +1


det

(

part + (Wxy)

partb + (mxy)

)

Strat

=

prod

(1 + a2Mx) minus

prod

(1 minus a2Mx)

prod


(1 minus a2m)


(


partτ +m

)



(2)1d and S


1d S(5)1d and S




Sfree =1

2

sum

xy

φxKxyφy +sum

xy


(1)free S

(4)free = S

(2)free S

(5)free S

(6)free = S

(3)free




1 minus (am2

)2)


= MTM = MTM = MTM


4a2∆2 = minus∆14





m

m(1)bos

m(1)ferm

m(5)ferm

m(6)ferm

00102030405060708091

0 0005 001 0015 002 0025

001020304050607080918899294969810




m

naive SLAC fermions


00102030405060708091

0005 001 0015 002 0025

0010203040506070809112131415161718


(4)ferm(xm) =

(

1 minus am

2

)

G(5)ferm

(

xm(

1 minus am

2

))



m(124)ferm (a) = m

(24)bos (a) asymp m

(

1 minus am

2

)


W (φ) =m

2φ2 +

g





bos(0) =




(5)bos(0) = 1678 plusmn 004 and m




m



00102030405060708091

0005 001 0015 002 0025 003

001020304050607080911212513135141451515516165




m



00102030405060708091

0005 001 0015 002 0025 003

00102030405060708091121251313514145151551616517




m



00102030405060708091

0005 001 0015 002 0025 003

00102030405060708091121251313514145151551616517














m


improved SLAC ferm


00102030405060708091

0 0005 001 0015 002 0025 003 0035

00102030405060708091121251313514145151551616517


int

D(φ ψ) eminusS+P


sum

y


(41)20


latti13e point x

R(1

)x

R(2)

x


0 5 10 15 20-0002-0001000010002




latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708090001-0002-0001000010002



latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708090001-0006-0004-00020000200040006







latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708090001-0006-0004-00020000200040006


2)〉 = 0






latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708091-002-0015-001-0005000050010015002



latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708091-002-0015-001-0005000050010015002




latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708091-0025-002-0015-001-00050000500100150020025




int

d2x(

2partφpartφ +1

2|W prime|2 + ψMψ

)


ψ =

(

ψ1

ψ2






2W primeε2





1

2a

(






1

a2

(

sum

micro=12




2

sum


sum

x

(


1

2|Wx|2

)







)

=

(

Wxy 2partxy

2partxy Wxy

)

Wxy =partWx

partφy (57)28


sum

xy




2εξx δ(1)ψ1

x = 0









ar


S(1)bos =

a2

2

sum

x

(


+ a2sum

x

(




2(amr) a2

sum


(1)bos =

a2

2

sum

xy


pmicro =1


2

asin

(apmicro2





2arp2 plusmn i|





iar



iar


S(2)bos =

a2

2

sum

x

(


+ a2sum

x

(



S(2)bos =

a2

2

sum

xy


microp = m2 +

p2 +(

12ar p2


radic

p2 +(

12arp2




r2 =4

3(75)31





(3)bos =

a2

2

sum

xy


ϕ =

(

ϕ1

ϕ2

) and ξ =

(

ξ1

ξ2

)


2mφ2 + 1


W prime(φ) = mϕ1 + u+ i(mϕ2 + v) (80)32



(n)0 +m)ϕ +

(

u

v


S(n)bos =

a2

2

sum

xy

(ϕx K(n)xy ϕy) +

a2

2

sum

x


sum

x

ux

(

(div ϕ)x minusar

2(∆ϕ1)x

)

minus a2sum

x

vx

(

(13url ϕ)x +ar

2(∆ϕ2)x

)

∆(2) = a2sum

x

ux

(

(div ϕ)x minusar

2(∆ϕ2)x

)

+ a2sum

x

vx

(


2(∆ϕ1)x

)

(84)∆(3) = a2

sum

x


x



1

2mφ2 +

1


m1minusloop = m(

1 minus g2

4πm2






m


002 004 006 008 01 012 01466577588599510105


35


partH


partφx (87)where

H =1

2

sum

x




prod

x

(

1 +m+ 3gφ2x

)


partM [φ]xypartφz




37


g=2g=1

g=05g=02g=01

R

p(R

)

50-5-10-15-20-25

25

2

15

1

05


quencheddynamical

t

Gbos(t

)

35302520151050

0045

004

0035

003

0025

002

0015

001

0005


39




(

G(x)

G(x+ a)

)

G = G(n)bos or G


G(x) =sum



Gferm(x) =sum

λ


λ (94)40




Gferm(x) =1

N

sum

k

e2πiNkx

2πiNk +m


latti13e point x

〈ψxψ

0〉


0 10 20 30 40 50 60-010010203040506070809


1e-09

1e-08

1e-07

1e-06

1e-05

1e-04

0001

001

01

1

0 20 40 60 80 100 120


latti13e point x

|〈ψxψ

0〉|



1

NtN2x

sum

tprime

sum

xxprime


42

andGferm(t) =

1

NtN2x

2sum

α=1

sum

tprime

sum

xxprime



1









1


1



Iε + Iπ =

int

BZ

d2k

(2π)2

1


Iε =

int

D

d2k

(2π)2

1

(k2 +m2)((k + q1)2 +m2) ((k + qnminus1)2 +m2)

Iπ =

int

BZD

d2k

(2π)2

1


le (Ca2)nminus1

int

|k|leradic

2πa

d2k

(2π)2

1


(

1 +2π2

a2m2

)



int

BZ

d2k

(2π)2



I primeε =

int

D

d2k

(2π)2


I primeπ =

int

BZD

d2k

(2π)2

Pmicro(k) P(k + ql)


leint

BZD

d2k

(2π)2


le(π

a

)l+1

(Ca2)nminus1

int

|k|leradic

2πa

d2k

(2π)2

1


(

1 +2π2

a2m2

)



iar

2γ3∆ with r2 =

4



47

internal linesq

k

propint

BZ

d2k

(2π)2



int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)+

int

BZ

d2k

(2π)2

m2

(P 2(k) +m2)(P 2(k minus q) +m2)


=

int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)

=

int

BZ

d2k

(2π)2


(k2 +m2)(P (k minus q)2 +m2)

minus 2Λsum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2

P micro(k minus q)



sum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2


le 2Cπ2

int minusΛ+q1

minusΛ

dk1

2π

int Λ

minusΛ

dk2

2π

1


=C

2

∣

∣

∣

∣

int Λ

minusΛ


)

ω(k2)minus1

∣

∣

∣

∣


le C

2

int Λ

minusΛ

dk2

∣

∣

∣

q1m2 + k2

2 minus Λq1 + Λ2

∣

∣







52

Introduction


Lattice models












Ward identities









Algorithmic aspects

Quantum Mechanics




Summary and outlook









a

2

sum

x

ξx(φ)2 + asum

xy

ψxpartξxpartφy


Ssusy =a

2

sum

x


sum

xy


= Snaive + asum

x



a

2

sum

x

ξx(φ)2 + asum

xy

ψxpartξypartφx


Ssusy =a

2

sum

x


sum

xy


= Snaive minus asum

x






sum






2∆xy +

partW prime(σx)


sum

x12

(




a






sum

xy


xy

vx(

part sxy +Wxy

)


sum

x


λ = minusar2

sum

xy

vx∆xyvy +sum

v



int

perbc

DψDψDφ eminusS =

int



(


partτ +m

)

=sinh (1

2

int β


sinh(β2m)



(

part + (Wxy)

partb +m1 )

Ito

=

prod




(

part + (Wxy)

partb +m1 )

Ito


e1

2βm2

det

(


partτ +m

) (30)sin13e lnprod




xy +1


det

(

part + (Wxy)

partb + (mxy)

)

Strat

=

prod

(1 + a2Mx) minus

prod

(1 minus a2Mx)

prod


(1 minus a2m)


(


partτ +m

)



(2)1d and S


1d S(5)1d and S




Sfree =1

2

sum

xy

φxKxyφy +sum

xy


(1)free S

(4)free = S

(2)free S

(5)free S

(6)free = S

(3)free




1 minus (am2

)2)


= MTM = MTM = MTM


4a2∆2 = minus∆14





m

m(1)bos

m(1)ferm

m(5)ferm

m(6)ferm

00102030405060708091

0 0005 001 0015 002 0025

001020304050607080918899294969810




m

naive SLAC fermions


00102030405060708091

0005 001 0015 002 0025

0010203040506070809112131415161718


(4)ferm(xm) =

(

1 minus am

2

)

G(5)ferm

(

xm(

1 minus am

2

))



m(124)ferm (a) = m

(24)bos (a) asymp m

(

1 minus am

2

)


W (φ) =m

2φ2 +

g





bos(0) =




(5)bos(0) = 1678 plusmn 004 and m




m



00102030405060708091

0005 001 0015 002 0025 003

001020304050607080911212513135141451515516165




m



00102030405060708091

0005 001 0015 002 0025 003

00102030405060708091121251313514145151551616517




m



00102030405060708091

0005 001 0015 002 0025 003

00102030405060708091121251313514145151551616517














m


improved SLAC ferm


00102030405060708091

0 0005 001 0015 002 0025 003 0035

00102030405060708091121251313514145151551616517


int

D(φ ψ) eminusS+P


sum

y


(41)20


latti13e point x

R(1

)x

R(2)

x


0 5 10 15 20-0002-0001000010002




latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708090001-0002-0001000010002



latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708090001-0006-0004-00020000200040006







latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708090001-0006-0004-00020000200040006


2)〉 = 0






latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708091-002-0015-001-0005000050010015002



latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708091-002-0015-001-0005000050010015002




latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708091-0025-002-0015-001-00050000500100150020025




int

d2x(

2partφpartφ +1

2|W prime|2 + ψMψ

)


ψ =

(

ψ1

ψ2






2W primeε2





1

2a

(






1

a2

(

sum

micro=12




2

sum


sum

x

(


1

2|Wx|2

)







)

=

(

Wxy 2partxy

2partxy Wxy

)

Wxy =partWx

partφy (57)28


sum

xy




2εξx δ(1)ψ1

x = 0









ar


S(1)bos =

a2

2

sum

x

(


+ a2sum

x

(




2(amr) a2

sum


(1)bos =

a2

2

sum

xy


pmicro =1


2

asin

(apmicro2





2arp2 plusmn i|





iar



iar


S(2)bos =

a2

2

sum

x

(


+ a2sum

x

(



S(2)bos =

a2

2

sum

xy


microp = m2 +

p2 +(

12ar p2


radic

p2 +(

12arp2




r2 =4

3(75)31





(3)bos =

a2

2

sum

xy


ϕ =

(

ϕ1

ϕ2

) and ξ =

(

ξ1

ξ2

)


2mφ2 + 1


W prime(φ) = mϕ1 + u+ i(mϕ2 + v) (80)32



(n)0 +m)ϕ +

(

u

v


S(n)bos =

a2

2

sum

xy

(ϕx K(n)xy ϕy) +

a2

2

sum

x


sum

x

ux

(

(div ϕ)x minusar

2(∆ϕ1)x

)

minus a2sum

x

vx

(

(13url ϕ)x +ar

2(∆ϕ2)x

)

∆(2) = a2sum

x

ux

(

(div ϕ)x minusar

2(∆ϕ2)x

)

+ a2sum

x

vx

(


2(∆ϕ1)x

)

(84)∆(3) = a2

sum

x


x



1

2mφ2 +

1


m1minusloop = m(

1 minus g2

4πm2






m


002 004 006 008 01 012 01466577588599510105


35


partH


partφx (87)where

H =1

2

sum

x




prod

x

(

1 +m+ 3gφ2x

)


partM [φ]xypartφz




37


g=2g=1

g=05g=02g=01

R

p(R

)

50-5-10-15-20-25

25

2

15

1

05


quencheddynamical

t

Gbos(t

)

35302520151050

0045

004

0035

003

0025

002

0015

001

0005


39




(

G(x)

G(x+ a)

)

G = G(n)bos or G


G(x) =sum



Gferm(x) =sum

λ


λ (94)40




Gferm(x) =1

N

sum

k

e2πiNkx

2πiNk +m


latti13e point x

〈ψxψ

0〉


0 10 20 30 40 50 60-010010203040506070809


1e-09

1e-08

1e-07

1e-06

1e-05

1e-04

0001

001

01

1

0 20 40 60 80 100 120


latti13e point x

|〈ψxψ

0〉|



1

NtN2x

sum

tprime

sum

xxprime


42

andGferm(t) =

1

NtN2x

2sum

α=1

sum

tprime

sum

xxprime



1









1


1



Iε + Iπ =

int

BZ

d2k

(2π)2

1


Iε =

int

D

d2k

(2π)2

1

(k2 +m2)((k + q1)2 +m2) ((k + qnminus1)2 +m2)

Iπ =

int

BZD

d2k

(2π)2

1


le (Ca2)nminus1

int

|k|leradic

2πa

d2k

(2π)2

1


(

1 +2π2

a2m2

)



int

BZ

d2k

(2π)2



I primeε =

int

D

d2k

(2π)2


I primeπ =

int

BZD

d2k

(2π)2

Pmicro(k) P(k + ql)


leint

BZD

d2k

(2π)2


le(π

a

)l+1

(Ca2)nminus1

int

|k|leradic

2πa

d2k

(2π)2

1


(

1 +2π2

a2m2

)



iar

2γ3∆ with r2 =

4



47

internal linesq

k

propint

BZ

d2k

(2π)2



int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)+

int

BZ

d2k

(2π)2

m2

(P 2(k) +m2)(P 2(k minus q) +m2)


=

int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)

=

int

BZ

d2k

(2π)2


(k2 +m2)(P (k minus q)2 +m2)

minus 2Λsum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2

P micro(k minus q)



sum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2


le 2Cπ2

int minusΛ+q1

minusΛ

dk1

2π

int Λ

minusΛ

dk2

2π

1


=C

2

∣

∣

∣

∣

int Λ

minusΛ


)

ω(k2)minus1

∣

∣

∣

∣


le C

2

int Λ

minusΛ

dk2

∣

∣

∣

q1m2 + k2

2 minus Λq1 + Λ2

∣

∣







52

Introduction


Lattice models












Ward identities









Algorithmic aspects

Quantum Mechanics




Summary and outlook






sum






2∆xy +

partW prime(σx)


sum

x12

(




a






sum

xy


xy

vx(

part sxy +Wxy

)


sum

x


λ = minusar2

sum

xy

vx∆xyvy +sum

v



int

perbc

DψDψDφ eminusS =

int



(


partτ +m

)

=sinh (1

2

int β


sinh(β2m)



(

part + (Wxy)

partb +m1 )

Ito

=

prod




(

part + (Wxy)

partb +m1 )

Ito


e1

2βm2

det

(


partτ +m

) (30)sin13e lnprod




xy +1


det

(

part + (Wxy)

partb + (mxy)

)

Strat

=

prod

(1 + a2Mx) minus

prod

(1 minus a2Mx)

prod


(1 minus a2m)


(


partτ +m

)



(2)1d and S


1d S(5)1d and S




Sfree =1

2

sum

xy

φxKxyφy +sum

xy


(1)free S

(4)free = S

(2)free S

(5)free S

(6)free = S

(3)free




1 minus (am2

)2)


= MTM = MTM = MTM


4a2∆2 = minus∆14





m

m(1)bos

m(1)ferm

m(5)ferm

m(6)ferm

00102030405060708091

0 0005 001 0015 002 0025

001020304050607080918899294969810




m

naive SLAC fermions


00102030405060708091

0005 001 0015 002 0025

0010203040506070809112131415161718


(4)ferm(xm) =

(

1 minus am

2

)

G(5)ferm

(

xm(

1 minus am

2

))



m(124)ferm (a) = m

(24)bos (a) asymp m

(

1 minus am

2

)


W (φ) =m

2φ2 +

g





bos(0) =




(5)bos(0) = 1678 plusmn 004 and m




m



00102030405060708091

0005 001 0015 002 0025 003

001020304050607080911212513135141451515516165




m



00102030405060708091

0005 001 0015 002 0025 003

00102030405060708091121251313514145151551616517




m



00102030405060708091

0005 001 0015 002 0025 003

00102030405060708091121251313514145151551616517














m


improved SLAC ferm


00102030405060708091

0 0005 001 0015 002 0025 003 0035

00102030405060708091121251313514145151551616517


int

D(φ ψ) eminusS+P


sum

y


(41)20


latti13e point x

R(1

)x

R(2)

x


0 5 10 15 20-0002-0001000010002




latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708090001-0002-0001000010002



latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708090001-0006-0004-00020000200040006







latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708090001-0006-0004-00020000200040006


2)〉 = 0






latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708091-002-0015-001-0005000050010015002



latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708091-002-0015-001-0005000050010015002




latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708091-0025-002-0015-001-00050000500100150020025




int

d2x(

2partφpartφ +1

2|W prime|2 + ψMψ

)


ψ =

(

ψ1

ψ2






2W primeε2





1

2a

(






1

a2

(

sum

micro=12




2

sum


sum

x

(


1

2|Wx|2

)







)

=

(

Wxy 2partxy

2partxy Wxy

)

Wxy =partWx

partφy (57)28


sum

xy




2εξx δ(1)ψ1

x = 0









ar


S(1)bos =

a2

2

sum

x

(


+ a2sum

x

(




2(amr) a2

sum


(1)bos =

a2

2

sum

xy


pmicro =1


2

asin

(apmicro2





2arp2 plusmn i|





iar



iar


S(2)bos =

a2

2

sum

x

(


+ a2sum

x

(



S(2)bos =

a2

2

sum

xy


microp = m2 +

p2 +(

12ar p2


radic

p2 +(

12arp2




r2 =4

3(75)31





(3)bos =

a2

2

sum

xy


ϕ =

(

ϕ1

ϕ2

) and ξ =

(

ξ1

ξ2

)


2mφ2 + 1


W prime(φ) = mϕ1 + u+ i(mϕ2 + v) (80)32



(n)0 +m)ϕ +

(

u

v


S(n)bos =

a2

2

sum

xy

(ϕx K(n)xy ϕy) +

a2

2

sum

x


sum

x

ux

(

(div ϕ)x minusar

2(∆ϕ1)x

)

minus a2sum

x

vx

(

(13url ϕ)x +ar

2(∆ϕ2)x

)

∆(2) = a2sum

x

ux

(

(div ϕ)x minusar

2(∆ϕ2)x

)

+ a2sum

x

vx

(


2(∆ϕ1)x

)

(84)∆(3) = a2

sum

x


x



1

2mφ2 +

1


m1minusloop = m(

1 minus g2

4πm2






m


002 004 006 008 01 012 01466577588599510105


35


partH


partφx (87)where

H =1

2

sum

x




prod

x

(

1 +m+ 3gφ2x

)


partM [φ]xypartφz




37


g=2g=1

g=05g=02g=01

R

p(R

)

50-5-10-15-20-25

25

2

15

1

05


quencheddynamical

t

Gbos(t

)

35302520151050

0045

004

0035

003

0025

002

0015

001

0005


39




(

G(x)

G(x+ a)

)

G = G(n)bos or G


G(x) =sum



Gferm(x) =sum

λ


λ (94)40




Gferm(x) =1

N

sum

k

e2πiNkx

2πiNk +m


latti13e point x

〈ψxψ

0〉


0 10 20 30 40 50 60-010010203040506070809


1e-09

1e-08

1e-07

1e-06

1e-05

1e-04

0001

001

01

1

0 20 40 60 80 100 120


latti13e point x

|〈ψxψ

0〉|



1

NtN2x

sum

tprime

sum

xxprime


42

andGferm(t) =

1

NtN2x

2sum

α=1

sum

tprime

sum

xxprime



1









1


1



Iε + Iπ =

int

BZ

d2k

(2π)2

1


Iε =

int

D

d2k

(2π)2

1

(k2 +m2)((k + q1)2 +m2) ((k + qnminus1)2 +m2)

Iπ =

int

BZD

d2k

(2π)2

1


le (Ca2)nminus1

int

|k|leradic

2πa

d2k

(2π)2

1


(

1 +2π2

a2m2

)



int

BZ

d2k

(2π)2



I primeε =

int

D

d2k

(2π)2


I primeπ =

int

BZD

d2k

(2π)2

Pmicro(k) P(k + ql)


leint

BZD

d2k

(2π)2


le(π

a

)l+1

(Ca2)nminus1

int

|k|leradic

2πa

d2k

(2π)2

1


(

1 +2π2

a2m2

)



iar

2γ3∆ with r2 =

4



47

internal linesq

k

propint

BZ

d2k

(2π)2



int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)+

int

BZ

d2k

(2π)2

m2

(P 2(k) +m2)(P 2(k minus q) +m2)


=

int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)

=

int

BZ

d2k

(2π)2


(k2 +m2)(P (k minus q)2 +m2)

minus 2Λsum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2

P micro(k minus q)



sum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2


le 2Cπ2

int minusΛ+q1

minusΛ

dk1

2π

int Λ

minusΛ

dk2

2π

1


=C

2

∣

∣

∣

∣

int Λ

minusΛ


)

ω(k2)minus1

∣

∣

∣

∣


le C

2

int Λ

minusΛ

dk2

∣

∣

∣

q1m2 + k2

2 minus Λq1 + Λ2

∣

∣







52

Introduction


Lattice models












Ward identities









Algorithmic aspects

Quantum Mechanics




Summary and outlook



a






sum

xy


xy

vx(

part sxy +Wxy

)


sum

x


λ = minusar2

sum

xy

vx∆xyvy +sum

v



int

perbc

DψDψDφ eminusS =

int



(


partτ +m

)

=sinh (1

2

int β


sinh(β2m)



(

part + (Wxy)

partb +m1 )

Ito

=

prod




(

part + (Wxy)

partb +m1 )

Ito


e1

2βm2

det

(


partτ +m

) (30)sin13e lnprod




xy +1


det

(

part + (Wxy)

partb + (mxy)

)

Strat

=

prod

(1 + a2Mx) minus

prod

(1 minus a2Mx)

prod


(1 minus a2m)


(


partτ +m

)



(2)1d and S


1d S(5)1d and S




Sfree =1

2

sum

xy

φxKxyφy +sum

xy


(1)free S

(4)free = S

(2)free S

(5)free S

(6)free = S

(3)free




1 minus (am2

)2)


= MTM = MTM = MTM


4a2∆2 = minus∆14





m

m(1)bos

m(1)ferm

m(5)ferm

m(6)ferm

00102030405060708091

0 0005 001 0015 002 0025

001020304050607080918899294969810




m

naive SLAC fermions


00102030405060708091

0005 001 0015 002 0025

0010203040506070809112131415161718


(4)ferm(xm) =

(

1 minus am

2

)

G(5)ferm

(

xm(

1 minus am

2

))



m(124)ferm (a) = m

(24)bos (a) asymp m

(

1 minus am

2

)


W (φ) =m

2φ2 +

g





bos(0) =




(5)bos(0) = 1678 plusmn 004 and m




m



00102030405060708091

0005 001 0015 002 0025 003

001020304050607080911212513135141451515516165




m



00102030405060708091

0005 001 0015 002 0025 003

00102030405060708091121251313514145151551616517




m



00102030405060708091

0005 001 0015 002 0025 003

00102030405060708091121251313514145151551616517














m


improved SLAC ferm


00102030405060708091

0 0005 001 0015 002 0025 003 0035

00102030405060708091121251313514145151551616517


int

D(φ ψ) eminusS+P


sum

y


(41)20


latti13e point x

R(1

)x

R(2)

x


0 5 10 15 20-0002-0001000010002




latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708090001-0002-0001000010002



latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708090001-0006-0004-00020000200040006







latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708090001-0006-0004-00020000200040006


2)〉 = 0






latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708091-002-0015-001-0005000050010015002



latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708091-002-0015-001-0005000050010015002




latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708091-0025-002-0015-001-00050000500100150020025




int

d2x(

2partφpartφ +1

2|W prime|2 + ψMψ

)


ψ =

(

ψ1

ψ2






2W primeε2





1

2a

(






1

a2

(

sum

micro=12




2

sum


sum

x

(


1

2|Wx|2

)







)

=

(

Wxy 2partxy

2partxy Wxy

)

Wxy =partWx

partφy (57)28


sum

xy




2εξx δ(1)ψ1

x = 0









ar


S(1)bos =

a2

2

sum

x

(


+ a2sum

x

(




2(amr) a2

sum


(1)bos =

a2

2

sum

xy


pmicro =1


2

asin

(apmicro2





2arp2 plusmn i|





iar



iar


S(2)bos =

a2

2

sum

x

(


+ a2sum

x

(



S(2)bos =

a2

2

sum

xy


microp = m2 +

p2 +(

12ar p2


radic

p2 +(

12arp2




r2 =4

3(75)31





(3)bos =

a2

2

sum

xy


ϕ =

(

ϕ1

ϕ2

) and ξ =

(

ξ1

ξ2

)


2mφ2 + 1


W prime(φ) = mϕ1 + u+ i(mϕ2 + v) (80)32



(n)0 +m)ϕ +

(

u

v


S(n)bos =

a2

2

sum

xy

(ϕx K(n)xy ϕy) +

a2

2

sum

x


sum

x

ux

(

(div ϕ)x minusar

2(∆ϕ1)x

)

minus a2sum

x

vx

(

(13url ϕ)x +ar

2(∆ϕ2)x

)

∆(2) = a2sum

x

ux

(

(div ϕ)x minusar

2(∆ϕ2)x

)

+ a2sum

x

vx

(


2(∆ϕ1)x

)

(84)∆(3) = a2

sum

x


x



1

2mφ2 +

1


m1minusloop = m(

1 minus g2

4πm2






m


002 004 006 008 01 012 01466577588599510105


35


partH


partφx (87)where

H =1

2

sum

x




prod

x

(

1 +m+ 3gφ2x

)


partM [φ]xypartφz




37


g=2g=1

g=05g=02g=01

R

p(R

)

50-5-10-15-20-25

25

2

15

1

05


quencheddynamical

t

Gbos(t

)

35302520151050

0045

004

0035

003

0025

002

0015

001

0005


39




(

G(x)

G(x+ a)

)

G = G(n)bos or G


G(x) =sum



Gferm(x) =sum

λ


λ (94)40




Gferm(x) =1

N

sum

k

e2πiNkx

2πiNk +m


latti13e point x

〈ψxψ

0〉


0 10 20 30 40 50 60-010010203040506070809


1e-09

1e-08

1e-07

1e-06

1e-05

1e-04

0001

001

01

1

0 20 40 60 80 100 120


latti13e point x

|〈ψxψ

0〉|



1

NtN2x

sum

tprime

sum

xxprime


42

andGferm(t) =

1

NtN2x

2sum

α=1

sum

tprime

sum

xxprime



1









1


1



Iε + Iπ =

int

BZ

d2k

(2π)2

1


Iε =

int

D

d2k

(2π)2

1

(k2 +m2)((k + q1)2 +m2) ((k + qnminus1)2 +m2)

Iπ =

int

BZD

d2k

(2π)2

1


le (Ca2)nminus1

int

|k|leradic

2πa

d2k

(2π)2

1


(

1 +2π2

a2m2

)



int

BZ

d2k

(2π)2



I primeε =

int

D

d2k

(2π)2


I primeπ =

int

BZD

d2k

(2π)2

Pmicro(k) P(k + ql)


leint

BZD

d2k

(2π)2


le(π

a

)l+1

(Ca2)nminus1

int

|k|leradic

2πa

d2k

(2π)2

1


(

1 +2π2

a2m2

)



iar

2γ3∆ with r2 =

4



47

internal linesq

k

propint

BZ

d2k

(2π)2



int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)+

int

BZ

d2k

(2π)2

m2

(P 2(k) +m2)(P 2(k minus q) +m2)


=

int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)

=

int

BZ

d2k

(2π)2


(k2 +m2)(P (k minus q)2 +m2)

minus 2Λsum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2

P micro(k minus q)



sum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2


le 2Cπ2

int minusΛ+q1

minusΛ

dk1

2π

int Λ

minusΛ

dk2

2π

1


=C

2

∣

∣

∣

∣

int Λ

minusΛ


)

ω(k2)minus1

∣

∣

∣

∣


le C

2

int Λ

minusΛ

dk2

∣

∣

∣

q1m2 + k2

2 minus Λq1 + Λ2

∣

∣







52

Introduction


Lattice models












Ward identities









Algorithmic aspects

Quantum Mechanics




Summary and outlook



sum

xy


xy

vx(

part sxy +Wxy

)


sum

x


λ = minusar2

sum

xy

vx∆xyvy +sum

v



int

perbc

DψDψDφ eminusS =

int



(


partτ +m

)

=sinh (1

2

int β


sinh(β2m)



(

part + (Wxy)

partb +m1 )

Ito

=

prod




(

part + (Wxy)

partb +m1 )

Ito


e1

2βm2

det

(


partτ +m

) (30)sin13e lnprod




xy +1


det

(

part + (Wxy)

partb + (mxy)

)

Strat

=

prod

(1 + a2Mx) minus

prod

(1 minus a2Mx)

prod


(1 minus a2m)


(


partτ +m

)



(2)1d and S


1d S(5)1d and S




Sfree =1

2

sum

xy

φxKxyφy +sum

xy


(1)free S

(4)free = S

(2)free S

(5)free S

(6)free = S

(3)free




1 minus (am2

)2)


= MTM = MTM = MTM


4a2∆2 = minus∆14





m

m(1)bos

m(1)ferm

m(5)ferm

m(6)ferm

00102030405060708091

0 0005 001 0015 002 0025

001020304050607080918899294969810




m

naive SLAC fermions


00102030405060708091

0005 001 0015 002 0025

0010203040506070809112131415161718


(4)ferm(xm) =

(

1 minus am

2

)

G(5)ferm

(

xm(

1 minus am

2

))



m(124)ferm (a) = m

(24)bos (a) asymp m

(

1 minus am

2

)


W (φ) =m

2φ2 +

g





bos(0) =




(5)bos(0) = 1678 plusmn 004 and m




m



00102030405060708091

0005 001 0015 002 0025 003

001020304050607080911212513135141451515516165




m



00102030405060708091

0005 001 0015 002 0025 003

00102030405060708091121251313514145151551616517




m



00102030405060708091

0005 001 0015 002 0025 003

00102030405060708091121251313514145151551616517














m


improved SLAC ferm


00102030405060708091

0 0005 001 0015 002 0025 003 0035

00102030405060708091121251313514145151551616517


int

D(φ ψ) eminusS+P


sum

y


(41)20


latti13e point x

R(1

)x

R(2)

x


0 5 10 15 20-0002-0001000010002




latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708090001-0002-0001000010002



latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708090001-0006-0004-00020000200040006







latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708090001-0006-0004-00020000200040006


2)〉 = 0






latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708091-002-0015-001-0005000050010015002



latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708091-002-0015-001-0005000050010015002




latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708091-0025-002-0015-001-00050000500100150020025




int

d2x(

2partφpartφ +1

2|W prime|2 + ψMψ

)


ψ =

(

ψ1

ψ2






2W primeε2





1

2a

(






1

a2

(

sum

micro=12




2

sum


sum

x

(


1

2|Wx|2

)







)

=

(

Wxy 2partxy

2partxy Wxy

)

Wxy =partWx

partφy (57)28


sum

xy




2εξx δ(1)ψ1

x = 0









ar


S(1)bos =

a2

2

sum

x

(


+ a2sum

x

(




2(amr) a2

sum


(1)bos =

a2

2

sum

xy


pmicro =1


2

asin

(apmicro2





2arp2 plusmn i|





iar



iar


S(2)bos =

a2

2

sum

x

(


+ a2sum

x

(



S(2)bos =

a2

2

sum

xy


microp = m2 +

p2 +(

12ar p2


radic

p2 +(

12arp2




r2 =4

3(75)31





(3)bos =

a2

2

sum

xy


ϕ =

(

ϕ1

ϕ2

) and ξ =

(

ξ1

ξ2

)


2mφ2 + 1


W prime(φ) = mϕ1 + u+ i(mϕ2 + v) (80)32



(n)0 +m)ϕ +

(

u

v


S(n)bos =

a2

2

sum

xy

(ϕx K(n)xy ϕy) +

a2

2

sum

x


sum

x

ux

(

(div ϕ)x minusar

2(∆ϕ1)x

)

minus a2sum

x

vx

(

(13url ϕ)x +ar

2(∆ϕ2)x

)

∆(2) = a2sum

x

ux

(

(div ϕ)x minusar

2(∆ϕ2)x

)

+ a2sum

x

vx

(


2(∆ϕ1)x

)

(84)∆(3) = a2

sum

x


x



1

2mφ2 +

1


m1minusloop = m(

1 minus g2

4πm2






m


002 004 006 008 01 012 01466577588599510105


35


partH


partφx (87)where

H =1

2

sum

x




prod

x

(

1 +m+ 3gφ2x

)


partM [φ]xypartφz




37


g=2g=1

g=05g=02g=01

R

p(R

)

50-5-10-15-20-25

25

2

15

1

05


quencheddynamical

t

Gbos(t

)

35302520151050

0045

004

0035

003

0025

002

0015

001

0005


39




(

G(x)

G(x+ a)

)

G = G(n)bos or G


G(x) =sum



Gferm(x) =sum

λ


λ (94)40




Gferm(x) =1

N

sum

k

e2πiNkx

2πiNk +m


latti13e point x

〈ψxψ

0〉


0 10 20 30 40 50 60-010010203040506070809


1e-09

1e-08

1e-07

1e-06

1e-05

1e-04

0001

001

01

1

0 20 40 60 80 100 120


latti13e point x

|〈ψxψ

0〉|



1

NtN2x

sum

tprime

sum

xxprime


42

andGferm(t) =

1

NtN2x

2sum

α=1

sum

tprime

sum

xxprime



1









1


1



Iε + Iπ =

int

BZ

d2k

(2π)2

1


Iε =

int

D

d2k

(2π)2

1

(k2 +m2)((k + q1)2 +m2) ((k + qnminus1)2 +m2)

Iπ =

int

BZD

d2k

(2π)2

1


le (Ca2)nminus1

int

|k|leradic

2πa

d2k

(2π)2

1


(

1 +2π2

a2m2

)



int

BZ

d2k

(2π)2



I primeε =

int

D

d2k

(2π)2


I primeπ =

int

BZD

d2k

(2π)2

Pmicro(k) P(k + ql)


leint

BZD

d2k

(2π)2


le(π

a

)l+1

(Ca2)nminus1

int

|k|leradic

2πa

d2k

(2π)2

1


(

1 +2π2

a2m2

)



iar

2γ3∆ with r2 =

4



47

internal linesq

k

propint

BZ

d2k

(2π)2



int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)+

int

BZ

d2k

(2π)2

m2

(P 2(k) +m2)(P 2(k minus q) +m2)


=

int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)

=

int

BZ

d2k

(2π)2


(k2 +m2)(P (k minus q)2 +m2)

minus 2Λsum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2

P micro(k minus q)



sum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2


le 2Cπ2

int minusΛ+q1

minusΛ

dk1

2π

int Λ

minusΛ

dk2

2π

1


=C

2

∣

∣

∣

∣

int Λ

minusΛ


)

ω(k2)minus1

∣

∣

∣

∣


le C

2

int Λ

minusΛ

dk2

∣

∣

∣

q1m2 + k2

2 minus Λq1 + Λ2

∣

∣







52

Introduction


Lattice models












Ward identities









Algorithmic aspects

Quantum Mechanics




Summary and outlook



(


partτ +m

)

=sinh (1

2

int β


sinh(β2m)



(

part + (Wxy)

partb +m1 )

Ito

=

prod




(

part + (Wxy)

partb +m1 )

Ito


e1

2βm2

det

(


partτ +m

) (30)sin13e lnprod




xy +1


det

(

part + (Wxy)

partb + (mxy)

)

Strat

=

prod

(1 + a2Mx) minus

prod

(1 minus a2Mx)

prod


(1 minus a2m)


(


partτ +m

)



(2)1d and S


1d S(5)1d and S




Sfree =1

2

sum

xy

φxKxyφy +sum

xy


(1)free S

(4)free = S

(2)free S

(5)free S

(6)free = S

(3)free




1 minus (am2

)2)


= MTM = MTM = MTM


4a2∆2 = minus∆14





m

m(1)bos

m(1)ferm

m(5)ferm

m(6)ferm

00102030405060708091

0 0005 001 0015 002 0025

001020304050607080918899294969810




m

naive SLAC fermions


00102030405060708091

0005 001 0015 002 0025

0010203040506070809112131415161718


(4)ferm(xm) =

(

1 minus am

2

)

G(5)ferm

(

xm(

1 minus am

2

))



m(124)ferm (a) = m

(24)bos (a) asymp m

(

1 minus am

2

)


W (φ) =m

2φ2 +

g





bos(0) =




(5)bos(0) = 1678 plusmn 004 and m




m



00102030405060708091

0005 001 0015 002 0025 003

001020304050607080911212513135141451515516165




m



00102030405060708091

0005 001 0015 002 0025 003

00102030405060708091121251313514145151551616517




m



00102030405060708091

0005 001 0015 002 0025 003

00102030405060708091121251313514145151551616517














m


improved SLAC ferm


00102030405060708091

0 0005 001 0015 002 0025 003 0035

00102030405060708091121251313514145151551616517


int

D(φ ψ) eminusS+P


sum

y


(41)20


latti13e point x

R(1

)x

R(2)

x


0 5 10 15 20-0002-0001000010002




latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708090001-0002-0001000010002



latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708090001-0006-0004-00020000200040006







latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708090001-0006-0004-00020000200040006


2)〉 = 0






latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708091-002-0015-001-0005000050010015002



latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708091-002-0015-001-0005000050010015002




latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708091-0025-002-0015-001-00050000500100150020025




int

d2x(

2partφpartφ +1

2|W prime|2 + ψMψ

)


ψ =

(

ψ1

ψ2






2W primeε2





1

2a

(






1

a2

(

sum

micro=12




2

sum


sum

x

(


1

2|Wx|2

)







)

=

(

Wxy 2partxy

2partxy Wxy

)

Wxy =partWx

partφy (57)28


sum

xy




2εξx δ(1)ψ1

x = 0









ar


S(1)bos =

a2

2

sum

x

(


+ a2sum

x

(




2(amr) a2

sum


(1)bos =

a2

2

sum

xy


pmicro =1


2

asin

(apmicro2





2arp2 plusmn i|





iar



iar


S(2)bos =

a2

2

sum

x

(


+ a2sum

x

(



S(2)bos =

a2

2

sum

xy


microp = m2 +

p2 +(

12ar p2


radic

p2 +(

12arp2




r2 =4

3(75)31





(3)bos =

a2

2

sum

xy


ϕ =

(

ϕ1

ϕ2

) and ξ =

(

ξ1

ξ2

)


2mφ2 + 1


W prime(φ) = mϕ1 + u+ i(mϕ2 + v) (80)32



(n)0 +m)ϕ +

(

u

v


S(n)bos =

a2

2

sum

xy

(ϕx K(n)xy ϕy) +

a2

2

sum

x


sum

x

ux

(

(div ϕ)x minusar

2(∆ϕ1)x

)

minus a2sum

x

vx

(

(13url ϕ)x +ar

2(∆ϕ2)x

)

∆(2) = a2sum

x

ux

(

(div ϕ)x minusar

2(∆ϕ2)x

)

+ a2sum

x

vx

(


2(∆ϕ1)x

)

(84)∆(3) = a2

sum

x


x



1

2mφ2 +

1


m1minusloop = m(

1 minus g2

4πm2






m


002 004 006 008 01 012 01466577588599510105


35


partH


partφx (87)where

H =1

2

sum

x




prod

x

(

1 +m+ 3gφ2x

)


partM [φ]xypartφz




37


g=2g=1

g=05g=02g=01

R

p(R

)

50-5-10-15-20-25

25

2

15

1

05


quencheddynamical

t

Gbos(t

)

35302520151050

0045

004

0035

003

0025

002

0015

001

0005


39




(

G(x)

G(x+ a)

)

G = G(n)bos or G


G(x) =sum



Gferm(x) =sum

λ


λ (94)40




Gferm(x) =1

N

sum

k

e2πiNkx

2πiNk +m


latti13e point x

〈ψxψ

0〉


0 10 20 30 40 50 60-010010203040506070809


1e-09

1e-08

1e-07

1e-06

1e-05

1e-04

0001

001

01

1

0 20 40 60 80 100 120


latti13e point x

|〈ψxψ

0〉|



1

NtN2x

sum

tprime

sum

xxprime


42

andGferm(t) =

1

NtN2x

2sum

α=1

sum

tprime

sum

xxprime



1









1


1



Iε + Iπ =

int

BZ

d2k

(2π)2

1


Iε =

int

D

d2k

(2π)2

1

(k2 +m2)((k + q1)2 +m2) ((k + qnminus1)2 +m2)

Iπ =

int

BZD

d2k

(2π)2

1


le (Ca2)nminus1

int

|k|leradic

2πa

d2k

(2π)2

1


(

1 +2π2

a2m2

)



int

BZ

d2k

(2π)2



I primeε =

int

D

d2k

(2π)2


I primeπ =

int

BZD

d2k

(2π)2

Pmicro(k) P(k + ql)


leint

BZD

d2k

(2π)2


le(π

a

)l+1

(Ca2)nminus1

int

|k|leradic

2πa

d2k

(2π)2

1


(

1 +2π2

a2m2

)



iar

2γ3∆ with r2 =

4



47

internal linesq

k

propint

BZ

d2k

(2π)2



int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)+

int

BZ

d2k

(2π)2

m2

(P 2(k) +m2)(P 2(k minus q) +m2)


=

int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)

=

int

BZ

d2k

(2π)2


(k2 +m2)(P (k minus q)2 +m2)

minus 2Λsum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2

P micro(k minus q)



sum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2


le 2Cπ2

int minusΛ+q1

minusΛ

dk1

2π

int Λ

minusΛ

dk2

2π

1


=C

2

∣

∣

∣

∣

int Λ

minusΛ


)

ω(k2)minus1

∣

∣

∣

∣


le C

2

int Λ

minusΛ

dk2

∣

∣

∣

q1m2 + k2

2 minus Λq1 + Λ2

∣

∣







52

Introduction


Lattice models












Ward identities









Algorithmic aspects

Quantum Mechanics




Summary and outlook



(2)1d and S


1d S(5)1d and S




Sfree =1

2

sum

xy

φxKxyφy +sum

xy


(1)free S

(4)free = S

(2)free S

(5)free S

(6)free = S

(3)free




1 minus (am2

)2)


= MTM = MTM = MTM


4a2∆2 = minus∆14





m

m(1)bos

m(1)ferm

m(5)ferm

m(6)ferm

00102030405060708091

0 0005 001 0015 002 0025

001020304050607080918899294969810




m

naive SLAC fermions


00102030405060708091

0005 001 0015 002 0025

0010203040506070809112131415161718


(4)ferm(xm) =

(

1 minus am

2

)

G(5)ferm

(

xm(

1 minus am

2

))



m(124)ferm (a) = m

(24)bos (a) asymp m

(

1 minus am

2

)


W (φ) =m

2φ2 +

g





bos(0) =




(5)bos(0) = 1678 plusmn 004 and m




m



00102030405060708091

0005 001 0015 002 0025 003

001020304050607080911212513135141451515516165




m



00102030405060708091

0005 001 0015 002 0025 003

00102030405060708091121251313514145151551616517




m



00102030405060708091

0005 001 0015 002 0025 003

00102030405060708091121251313514145151551616517














m


improved SLAC ferm


00102030405060708091

0 0005 001 0015 002 0025 003 0035

00102030405060708091121251313514145151551616517


int

D(φ ψ) eminusS+P


sum

y


(41)20


latti13e point x

R(1

)x

R(2)

x


0 5 10 15 20-0002-0001000010002




latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708090001-0002-0001000010002



latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708090001-0006-0004-00020000200040006







latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708090001-0006-0004-00020000200040006


2)〉 = 0






latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708091-002-0015-001-0005000050010015002



latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708091-002-0015-001-0005000050010015002




latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708091-0025-002-0015-001-00050000500100150020025




int

d2x(

2partφpartφ +1

2|W prime|2 + ψMψ

)


ψ =

(

ψ1

ψ2






2W primeε2





1

2a

(






1

a2

(

sum

micro=12




2

sum


sum

x

(


1

2|Wx|2

)







)

=

(

Wxy 2partxy

2partxy Wxy

)

Wxy =partWx

partφy (57)28


sum

xy




2εξx δ(1)ψ1

x = 0









ar


S(1)bos =

a2

2

sum

x

(


+ a2sum

x

(




2(amr) a2

sum


(1)bos =

a2

2

sum

xy


pmicro =1


2

asin

(apmicro2





2arp2 plusmn i|





iar



iar


S(2)bos =

a2

2

sum

x

(


+ a2sum

x

(



S(2)bos =

a2

2

sum

xy


microp = m2 +

p2 +(

12ar p2


radic

p2 +(

12arp2




r2 =4

3(75)31





(3)bos =

a2

2

sum

xy


ϕ =

(

ϕ1

ϕ2

) and ξ =

(

ξ1

ξ2

)


2mφ2 + 1


W prime(φ) = mϕ1 + u+ i(mϕ2 + v) (80)32



(n)0 +m)ϕ +

(

u

v


S(n)bos =

a2

2

sum

xy

(ϕx K(n)xy ϕy) +

a2

2

sum

x


sum

x

ux

(

(div ϕ)x minusar

2(∆ϕ1)x

)

minus a2sum

x

vx

(

(13url ϕ)x +ar

2(∆ϕ2)x

)

∆(2) = a2sum

x

ux

(

(div ϕ)x minusar

2(∆ϕ2)x

)

+ a2sum

x

vx

(


2(∆ϕ1)x

)

(84)∆(3) = a2

sum

x


x



1

2mφ2 +

1


m1minusloop = m(

1 minus g2

4πm2






m


002 004 006 008 01 012 01466577588599510105


35


partH


partφx (87)where

H =1

2

sum

x




prod

x

(

1 +m+ 3gφ2x

)


partM [φ]xypartφz




37


g=2g=1

g=05g=02g=01

R

p(R

)

50-5-10-15-20-25

25

2

15

1

05


quencheddynamical

t

Gbos(t

)

35302520151050

0045

004

0035

003

0025

002

0015

001

0005


39




(

G(x)

G(x+ a)

)

G = G(n)bos or G


G(x) =sum



Gferm(x) =sum

λ


λ (94)40




Gferm(x) =1

N

sum

k

e2πiNkx

2πiNk +m


latti13e point x

〈ψxψ

0〉


0 10 20 30 40 50 60-010010203040506070809


1e-09

1e-08

1e-07

1e-06

1e-05

1e-04

0001

001

01

1

0 20 40 60 80 100 120


latti13e point x

|〈ψxψ

0〉|



1

NtN2x

sum

tprime

sum

xxprime


42

andGferm(t) =

1

NtN2x

2sum

α=1

sum

tprime

sum

xxprime



1









1


1



Iε + Iπ =

int

BZ

d2k

(2π)2

1


Iε =

int

D

d2k

(2π)2

1

(k2 +m2)((k + q1)2 +m2) ((k + qnminus1)2 +m2)

Iπ =

int

BZD

d2k

(2π)2

1


le (Ca2)nminus1

int

|k|leradic

2πa

d2k

(2π)2

1


(

1 +2π2

a2m2

)



int

BZ

d2k

(2π)2



I primeε =

int

D

d2k

(2π)2


I primeπ =

int

BZD

d2k

(2π)2

Pmicro(k) P(k + ql)


leint

BZD

d2k

(2π)2


le(π

a

)l+1

(Ca2)nminus1

int

|k|leradic

2πa

d2k

(2π)2

1


(

1 +2π2

a2m2

)



iar

2γ3∆ with r2 =

4



47

internal linesq

k

propint

BZ

d2k

(2π)2



int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)+

int

BZ

d2k

(2π)2

m2

(P 2(k) +m2)(P 2(k minus q) +m2)


=

int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)

=

int

BZ

d2k

(2π)2


(k2 +m2)(P (k minus q)2 +m2)

minus 2Λsum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2

P micro(k minus q)



sum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2


le 2Cπ2

int minusΛ+q1

minusΛ

dk1

2π

int Λ

minusΛ

dk2

2π

1


=C

2

∣

∣

∣

∣

int Λ

minusΛ


)

ω(k2)minus1

∣

∣

∣

∣


le C

2

int Λ

minusΛ

dk2

∣

∣

∣

q1m2 + k2

2 minus Λq1 + Λ2

∣

∣







52

Introduction


Lattice models












Ward identities









Algorithmic aspects

Quantum Mechanics




Summary and outlook






m

m(1)bos

m(1)ferm

m(5)ferm

m(6)ferm

00102030405060708091

0 0005 001 0015 002 0025

001020304050607080918899294969810




m

naive SLAC fermions


00102030405060708091

0005 001 0015 002 0025

0010203040506070809112131415161718


(4)ferm(xm) =

(

1 minus am

2

)

G(5)ferm

(

xm(

1 minus am

2

))



m(124)ferm (a) = m

(24)bos (a) asymp m

(

1 minus am

2

)


W (φ) =m

2φ2 +

g





bos(0) =




(5)bos(0) = 1678 plusmn 004 and m




m



00102030405060708091

0005 001 0015 002 0025 003

001020304050607080911212513135141451515516165




m



00102030405060708091

0005 001 0015 002 0025 003

00102030405060708091121251313514145151551616517




m



00102030405060708091

0005 001 0015 002 0025 003

00102030405060708091121251313514145151551616517














m


improved SLAC ferm


00102030405060708091

0 0005 001 0015 002 0025 003 0035

00102030405060708091121251313514145151551616517


int

D(φ ψ) eminusS+P


sum

y


(41)20


latti13e point x

R(1

)x

R(2)

x


0 5 10 15 20-0002-0001000010002




latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708090001-0002-0001000010002



latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708090001-0006-0004-00020000200040006







latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708090001-0006-0004-00020000200040006


2)〉 = 0






latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708091-002-0015-001-0005000050010015002



latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708091-002-0015-001-0005000050010015002




latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708091-0025-002-0015-001-00050000500100150020025




int

d2x(

2partφpartφ +1

2|W prime|2 + ψMψ

)


ψ =

(

ψ1

ψ2






2W primeε2





1

2a

(






1

a2

(

sum

micro=12




2

sum


sum

x

(


1

2|Wx|2

)







)

=

(

Wxy 2partxy

2partxy Wxy

)

Wxy =partWx

partφy (57)28


sum

xy




2εξx δ(1)ψ1

x = 0









ar


S(1)bos =

a2

2

sum

x

(


+ a2sum

x

(




2(amr) a2

sum


(1)bos =

a2

2

sum

xy


pmicro =1


2

asin

(apmicro2





2arp2 plusmn i|





iar



iar


S(2)bos =

a2

2

sum

x

(


+ a2sum

x

(



S(2)bos =

a2

2

sum

xy


microp = m2 +

p2 +(

12ar p2


radic

p2 +(

12arp2




r2 =4

3(75)31





(3)bos =

a2

2

sum

xy


ϕ =

(

ϕ1

ϕ2

) and ξ =

(

ξ1

ξ2

)


2mφ2 + 1


W prime(φ) = mϕ1 + u+ i(mϕ2 + v) (80)32



(n)0 +m)ϕ +

(

u

v


S(n)bos =

a2

2

sum

xy

(ϕx K(n)xy ϕy) +

a2

2

sum

x


sum

x

ux

(

(div ϕ)x minusar

2(∆ϕ1)x

)

minus a2sum

x

vx

(

(13url ϕ)x +ar

2(∆ϕ2)x

)

∆(2) = a2sum

x

ux

(

(div ϕ)x minusar

2(∆ϕ2)x

)

+ a2sum

x

vx

(


2(∆ϕ1)x

)

(84)∆(3) = a2

sum

x


x



1

2mφ2 +

1


m1minusloop = m(

1 minus g2

4πm2






m


002 004 006 008 01 012 01466577588599510105


35


partH


partφx (87)where

H =1

2

sum

x




prod

x

(

1 +m+ 3gφ2x

)


partM [φ]xypartφz




37


g=2g=1

g=05g=02g=01

R

p(R

)

50-5-10-15-20-25

25

2

15

1

05


quencheddynamical

t

Gbos(t

)

35302520151050

0045

004

0035

003

0025

002

0015

001

0005


39




(

G(x)

G(x+ a)

)

G = G(n)bos or G


G(x) =sum



Gferm(x) =sum

λ


λ (94)40




Gferm(x) =1

N

sum

k

e2πiNkx

2πiNk +m


latti13e point x

〈ψxψ

0〉


0 10 20 30 40 50 60-010010203040506070809


1e-09

1e-08

1e-07

1e-06

1e-05

1e-04

0001

001

01

1

0 20 40 60 80 100 120


latti13e point x

|〈ψxψ

0〉|



1

NtN2x

sum

tprime

sum

xxprime


42

andGferm(t) =

1

NtN2x

2sum

α=1

sum

tprime

sum

xxprime



1









1


1



Iε + Iπ =

int

BZ

d2k

(2π)2

1


Iε =

int

D

d2k

(2π)2

1

(k2 +m2)((k + q1)2 +m2) ((k + qnminus1)2 +m2)

Iπ =

int

BZD

d2k

(2π)2

1


le (Ca2)nminus1

int

|k|leradic

2πa

d2k

(2π)2

1


(

1 +2π2

a2m2

)



int

BZ

d2k

(2π)2



I primeε =

int

D

d2k

(2π)2


I primeπ =

int

BZD

d2k

(2π)2

Pmicro(k) P(k + ql)


leint

BZD

d2k

(2π)2


le(π

a

)l+1

(Ca2)nminus1

int

|k|leradic

2πa

d2k

(2π)2

1


(

1 +2π2

a2m2

)



iar

2γ3∆ with r2 =

4



47

internal linesq

k

propint

BZ

d2k

(2π)2



int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)+

int

BZ

d2k

(2π)2

m2

(P 2(k) +m2)(P 2(k minus q) +m2)


=

int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)

=

int

BZ

d2k

(2π)2


(k2 +m2)(P (k minus q)2 +m2)

minus 2Λsum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2

P micro(k minus q)



sum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2


le 2Cπ2

int minusΛ+q1

minusΛ

dk1

2π

int Λ

minusΛ

dk2

2π

1


=C

2

∣

∣

∣

∣

int Λ

minusΛ


)

ω(k2)minus1

∣

∣

∣

∣


le C

2

int Λ

minusΛ

dk2

∣

∣

∣

q1m2 + k2

2 minus Λq1 + Λ2

∣

∣







52

Introduction


Lattice models












Ward identities









Algorithmic aspects

Quantum Mechanics




Summary and outlook




m

naive SLAC fermions


00102030405060708091

0005 001 0015 002 0025

0010203040506070809112131415161718


(4)ferm(xm) =

(

1 minus am

2

)

G(5)ferm

(

xm(

1 minus am

2

))



m(124)ferm (a) = m

(24)bos (a) asymp m

(

1 minus am

2

)


W (φ) =m

2φ2 +

g





bos(0) =




(5)bos(0) = 1678 plusmn 004 and m




m



00102030405060708091

0005 001 0015 002 0025 003

001020304050607080911212513135141451515516165




m



00102030405060708091

0005 001 0015 002 0025 003

00102030405060708091121251313514145151551616517




m



00102030405060708091

0005 001 0015 002 0025 003

00102030405060708091121251313514145151551616517














m


improved SLAC ferm


00102030405060708091

0 0005 001 0015 002 0025 003 0035

00102030405060708091121251313514145151551616517


int

D(φ ψ) eminusS+P


sum

y


(41)20


latti13e point x

R(1

)x

R(2)

x


0 5 10 15 20-0002-0001000010002




latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708090001-0002-0001000010002



latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708090001-0006-0004-00020000200040006







latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708090001-0006-0004-00020000200040006


2)〉 = 0






latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708091-002-0015-001-0005000050010015002



latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708091-002-0015-001-0005000050010015002




latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708091-0025-002-0015-001-00050000500100150020025




int

d2x(

2partφpartφ +1

2|W prime|2 + ψMψ

)


ψ =

(

ψ1

ψ2






2W primeε2





1

2a

(






1

a2

(

sum

micro=12




2

sum


sum

x

(


1

2|Wx|2

)







)

=

(

Wxy 2partxy

2partxy Wxy

)

Wxy =partWx

partφy (57)28


sum

xy




2εξx δ(1)ψ1

x = 0









ar


S(1)bos =

a2

2

sum

x

(


+ a2sum

x

(




2(amr) a2

sum


(1)bos =

a2

2

sum

xy


pmicro =1


2

asin

(apmicro2





2arp2 plusmn i|





iar



iar


S(2)bos =

a2

2

sum

x

(


+ a2sum

x

(



S(2)bos =

a2

2

sum

xy


microp = m2 +

p2 +(

12ar p2


radic

p2 +(

12arp2




r2 =4

3(75)31





(3)bos =

a2

2

sum

xy


ϕ =

(

ϕ1

ϕ2

) and ξ =

(

ξ1

ξ2

)


2mφ2 + 1


W prime(φ) = mϕ1 + u+ i(mϕ2 + v) (80)32



(n)0 +m)ϕ +

(

u

v


S(n)bos =

a2

2

sum

xy

(ϕx K(n)xy ϕy) +

a2

2

sum

x


sum

x

ux

(

(div ϕ)x minusar

2(∆ϕ1)x

)

minus a2sum

x

vx

(

(13url ϕ)x +ar

2(∆ϕ2)x

)

∆(2) = a2sum

x

ux

(

(div ϕ)x minusar

2(∆ϕ2)x

)

+ a2sum

x

vx

(


2(∆ϕ1)x

)

(84)∆(3) = a2

sum

x


x



1

2mφ2 +

1


m1minusloop = m(

1 minus g2

4πm2






m


002 004 006 008 01 012 01466577588599510105


35


partH


partφx (87)where

H =1

2

sum

x




prod

x

(

1 +m+ 3gφ2x

)


partM [φ]xypartφz




37


g=2g=1

g=05g=02g=01

R

p(R

)

50-5-10-15-20-25

25

2

15

1

05


quencheddynamical

t

Gbos(t

)

35302520151050

0045

004

0035

003

0025

002

0015

001

0005


39




(

G(x)

G(x+ a)

)

G = G(n)bos or G


G(x) =sum



Gferm(x) =sum

λ


λ (94)40




Gferm(x) =1

N

sum

k

e2πiNkx

2πiNk +m


latti13e point x

〈ψxψ

0〉


0 10 20 30 40 50 60-010010203040506070809


1e-09

1e-08

1e-07

1e-06

1e-05

1e-04

0001

001

01

1

0 20 40 60 80 100 120


latti13e point x

|〈ψxψ

0〉|



1

NtN2x

sum

tprime

sum

xxprime


42

andGferm(t) =

1

NtN2x

2sum

α=1

sum

tprime

sum

xxprime



1









1


1



Iε + Iπ =

int

BZ

d2k

(2π)2

1


Iε =

int

D

d2k

(2π)2

1

(k2 +m2)((k + q1)2 +m2) ((k + qnminus1)2 +m2)

Iπ =

int

BZD

d2k

(2π)2

1


le (Ca2)nminus1

int

|k|leradic

2πa

d2k

(2π)2

1


(

1 +2π2

a2m2

)



int

BZ

d2k

(2π)2



I primeε =

int

D

d2k

(2π)2


I primeπ =

int

BZD

d2k

(2π)2

Pmicro(k) P(k + ql)


leint

BZD

d2k

(2π)2


le(π

a

)l+1

(Ca2)nminus1

int

|k|leradic

2πa

d2k

(2π)2

1


(

1 +2π2

a2m2

)



iar

2γ3∆ with r2 =

4



47

internal linesq

k

propint

BZ

d2k

(2π)2



int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)+

int

BZ

d2k

(2π)2

m2

(P 2(k) +m2)(P 2(k minus q) +m2)


=

int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)

=

int

BZ

d2k

(2π)2


(k2 +m2)(P (k minus q)2 +m2)

minus 2Λsum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2

P micro(k minus q)



sum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2


le 2Cπ2

int minusΛ+q1

minusΛ

dk1

2π

int Λ

minusΛ

dk2

2π

1


=C

2

∣

∣

∣

∣

int Λ

minusΛ


)

ω(k2)minus1

∣

∣

∣

∣


le C

2

int Λ

minusΛ

dk2

∣

∣

∣

q1m2 + k2

2 minus Λq1 + Λ2

∣

∣







52

Introduction


Lattice models












Ward identities









Algorithmic aspects

Quantum Mechanics




Summary and outlook





bos(0) =




(5)bos(0) = 1678 plusmn 004 and m




m



00102030405060708091

0005 001 0015 002 0025 003

001020304050607080911212513135141451515516165




m



00102030405060708091

0005 001 0015 002 0025 003

00102030405060708091121251313514145151551616517




m



00102030405060708091

0005 001 0015 002 0025 003

00102030405060708091121251313514145151551616517














m


improved SLAC ferm


00102030405060708091

0 0005 001 0015 002 0025 003 0035

00102030405060708091121251313514145151551616517


int

D(φ ψ) eminusS+P


sum

y


(41)20


latti13e point x

R(1

)x

R(2)

x


0 5 10 15 20-0002-0001000010002




latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708090001-0002-0001000010002



latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708090001-0006-0004-00020000200040006







latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708090001-0006-0004-00020000200040006


2)〉 = 0






latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708091-002-0015-001-0005000050010015002



latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708091-002-0015-001-0005000050010015002




latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708091-0025-002-0015-001-00050000500100150020025




int

d2x(

2partφpartφ +1

2|W prime|2 + ψMψ

)


ψ =

(

ψ1

ψ2






2W primeε2





1

2a

(






1

a2

(

sum

micro=12




2

sum


sum

x

(


1

2|Wx|2

)







)

=

(

Wxy 2partxy

2partxy Wxy

)

Wxy =partWx

partφy (57)28


sum

xy




2εξx δ(1)ψ1

x = 0









ar


S(1)bos =

a2

2

sum

x

(


+ a2sum

x

(




2(amr) a2

sum


(1)bos =

a2

2

sum

xy


pmicro =1


2

asin

(apmicro2





2arp2 plusmn i|





iar



iar


S(2)bos =

a2

2

sum

x

(


+ a2sum

x

(



S(2)bos =

a2

2

sum

xy


microp = m2 +

p2 +(

12ar p2


radic

p2 +(

12arp2




r2 =4

3(75)31





(3)bos =

a2

2

sum

xy


ϕ =

(

ϕ1

ϕ2

) and ξ =

(

ξ1

ξ2

)


2mφ2 + 1


W prime(φ) = mϕ1 + u+ i(mϕ2 + v) (80)32



(n)0 +m)ϕ +

(

u

v


S(n)bos =

a2

2

sum

xy

(ϕx K(n)xy ϕy) +

a2

2

sum

x


sum

x

ux

(

(div ϕ)x minusar

2(∆ϕ1)x

)

minus a2sum

x

vx

(

(13url ϕ)x +ar

2(∆ϕ2)x

)

∆(2) = a2sum

x

ux

(

(div ϕ)x minusar

2(∆ϕ2)x

)

+ a2sum

x

vx

(


2(∆ϕ1)x

)

(84)∆(3) = a2

sum

x


x



1

2mφ2 +

1


m1minusloop = m(

1 minus g2

4πm2






m


002 004 006 008 01 012 01466577588599510105


35


partH


partφx (87)where

H =1

2

sum

x




prod

x

(

1 +m+ 3gφ2x

)


partM [φ]xypartφz




37


g=2g=1

g=05g=02g=01

R

p(R

)

50-5-10-15-20-25

25

2

15

1

05


quencheddynamical

t

Gbos(t

)

35302520151050

0045

004

0035

003

0025

002

0015

001

0005


39




(

G(x)

G(x+ a)

)

G = G(n)bos or G


G(x) =sum



Gferm(x) =sum

λ


λ (94)40




Gferm(x) =1

N

sum

k

e2πiNkx

2πiNk +m


latti13e point x

〈ψxψ

0〉


0 10 20 30 40 50 60-010010203040506070809


1e-09

1e-08

1e-07

1e-06

1e-05

1e-04

0001

001

01

1

0 20 40 60 80 100 120


latti13e point x

|〈ψxψ

0〉|



1

NtN2x

sum

tprime

sum

xxprime


42

andGferm(t) =

1

NtN2x

2sum

α=1

sum

tprime

sum

xxprime



1









1


1



Iε + Iπ =

int

BZ

d2k

(2π)2

1


Iε =

int

D

d2k

(2π)2

1

(k2 +m2)((k + q1)2 +m2) ((k + qnminus1)2 +m2)

Iπ =

int

BZD

d2k

(2π)2

1


le (Ca2)nminus1

int

|k|leradic

2πa

d2k

(2π)2

1


(

1 +2π2

a2m2

)



int

BZ

d2k

(2π)2



I primeε =

int

D

d2k

(2π)2


I primeπ =

int

BZD

d2k

(2π)2

Pmicro(k) P(k + ql)


leint

BZD

d2k

(2π)2


le(π

a

)l+1

(Ca2)nminus1

int

|k|leradic

2πa

d2k

(2π)2

1


(

1 +2π2

a2m2

)



iar

2γ3∆ with r2 =

4



47

internal linesq

k

propint

BZ

d2k

(2π)2



int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)+

int

BZ

d2k

(2π)2

m2

(P 2(k) +m2)(P 2(k minus q) +m2)


=

int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)

=

int

BZ

d2k

(2π)2


(k2 +m2)(P (k minus q)2 +m2)

minus 2Λsum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2

P micro(k minus q)



sum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2


le 2Cπ2

int minusΛ+q1

minusΛ

dk1

2π

int Λ

minusΛ

dk2

2π

1


=C

2

∣

∣

∣

∣

int Λ

minusΛ


)

ω(k2)minus1

∣

∣

∣

∣


le C

2

int Λ

minusΛ

dk2

∣

∣

∣

q1m2 + k2

2 minus Λq1 + Λ2

∣

∣







52

Introduction


Lattice models












Ward identities









Algorithmic aspects

Quantum Mechanics




Summary and outlook




m



00102030405060708091

0005 001 0015 002 0025 003

001020304050607080911212513135141451515516165




m



00102030405060708091

0005 001 0015 002 0025 003

00102030405060708091121251313514145151551616517




m



00102030405060708091

0005 001 0015 002 0025 003

00102030405060708091121251313514145151551616517














m


improved SLAC ferm


00102030405060708091

0 0005 001 0015 002 0025 003 0035

00102030405060708091121251313514145151551616517


int

D(φ ψ) eminusS+P


sum

y


(41)20


latti13e point x

R(1

)x

R(2)

x


0 5 10 15 20-0002-0001000010002




latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708090001-0002-0001000010002



latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708090001-0006-0004-00020000200040006







latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708090001-0006-0004-00020000200040006


2)〉 = 0






latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708091-002-0015-001-0005000050010015002



latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708091-002-0015-001-0005000050010015002




latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708091-0025-002-0015-001-00050000500100150020025




int

d2x(

2partφpartφ +1

2|W prime|2 + ψMψ

)


ψ =

(

ψ1

ψ2






2W primeε2





1

2a

(






1

a2

(

sum

micro=12




2

sum


sum

x

(


1

2|Wx|2

)







)

=

(

Wxy 2partxy

2partxy Wxy

)

Wxy =partWx

partφy (57)28


sum

xy




2εξx δ(1)ψ1

x = 0









ar


S(1)bos =

a2

2

sum

x

(


+ a2sum

x

(




2(amr) a2

sum


(1)bos =

a2

2

sum

xy


pmicro =1


2

asin

(apmicro2





2arp2 plusmn i|





iar



iar


S(2)bos =

a2

2

sum

x

(


+ a2sum

x

(



S(2)bos =

a2

2

sum

xy


microp = m2 +

p2 +(

12ar p2


radic

p2 +(

12arp2




r2 =4

3(75)31





(3)bos =

a2

2

sum

xy


ϕ =

(

ϕ1

ϕ2

) and ξ =

(

ξ1

ξ2

)


2mφ2 + 1


W prime(φ) = mϕ1 + u+ i(mϕ2 + v) (80)32



(n)0 +m)ϕ +

(

u

v


S(n)bos =

a2

2

sum

xy

(ϕx K(n)xy ϕy) +

a2

2

sum

x


sum

x

ux

(

(div ϕ)x minusar

2(∆ϕ1)x

)

minus a2sum

x

vx

(

(13url ϕ)x +ar

2(∆ϕ2)x

)

∆(2) = a2sum

x

ux

(

(div ϕ)x minusar

2(∆ϕ2)x

)

+ a2sum

x

vx

(


2(∆ϕ1)x

)

(84)∆(3) = a2

sum

x


x



1

2mφ2 +

1


m1minusloop = m(

1 minus g2

4πm2






m


002 004 006 008 01 012 01466577588599510105


35


partH


partφx (87)where

H =1

2

sum

x




prod

x

(

1 +m+ 3gφ2x

)


partM [φ]xypartφz




37


g=2g=1

g=05g=02g=01

R

p(R

)

50-5-10-15-20-25

25

2

15

1

05


quencheddynamical

t

Gbos(t

)

35302520151050

0045

004

0035

003

0025

002

0015

001

0005


39




(

G(x)

G(x+ a)

)

G = G(n)bos or G


G(x) =sum



Gferm(x) =sum

λ


λ (94)40




Gferm(x) =1

N

sum

k

e2πiNkx

2πiNk +m


latti13e point x

〈ψxψ

0〉


0 10 20 30 40 50 60-010010203040506070809


1e-09

1e-08

1e-07

1e-06

1e-05

1e-04

0001

001

01

1

0 20 40 60 80 100 120


latti13e point x

|〈ψxψ

0〉|



1

NtN2x

sum

tprime

sum

xxprime


42

andGferm(t) =

1

NtN2x

2sum

α=1

sum

tprime

sum

xxprime



1









1


1



Iε + Iπ =

int

BZ

d2k

(2π)2

1


Iε =

int

D

d2k

(2π)2

1

(k2 +m2)((k + q1)2 +m2) ((k + qnminus1)2 +m2)

Iπ =

int

BZD

d2k

(2π)2

1


le (Ca2)nminus1

int

|k|leradic

2πa

d2k

(2π)2

1


(

1 +2π2

a2m2

)



int

BZ

d2k

(2π)2



I primeε =

int

D

d2k

(2π)2


I primeπ =

int

BZD

d2k

(2π)2

Pmicro(k) P(k + ql)


leint

BZD

d2k

(2π)2


le(π

a

)l+1

(Ca2)nminus1

int

|k|leradic

2πa

d2k

(2π)2

1


(

1 +2π2

a2m2

)



iar

2γ3∆ with r2 =

4



47

internal linesq

k

propint

BZ

d2k

(2π)2



int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)+

int

BZ

d2k

(2π)2

m2

(P 2(k) +m2)(P 2(k minus q) +m2)


=

int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)

=

int

BZ

d2k

(2π)2


(k2 +m2)(P (k minus q)2 +m2)

minus 2Λsum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2

P micro(k minus q)



sum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2


le 2Cπ2

int minusΛ+q1

minusΛ

dk1

2π

int Λ

minusΛ

dk2

2π

1


=C

2

∣

∣

∣

∣

int Λ

minusΛ


)

ω(k2)minus1

∣

∣

∣

∣


le C

2

int Λ

minusΛ

dk2

∣

∣

∣

q1m2 + k2

2 minus Λq1 + Λ2

∣

∣







52

Introduction


Lattice models












Ward identities









Algorithmic aspects

Quantum Mechanics




Summary and outlook




m



00102030405060708091

0005 001 0015 002 0025 003

00102030405060708091121251313514145151551616517














m


improved SLAC ferm


00102030405060708091

0 0005 001 0015 002 0025 003 0035

00102030405060708091121251313514145151551616517


int

D(φ ψ) eminusS+P


sum

y


(41)20


latti13e point x

R(1

)x

R(2)

x


0 5 10 15 20-0002-0001000010002




latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708090001-0002-0001000010002



latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708090001-0006-0004-00020000200040006







latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708090001-0006-0004-00020000200040006


2)〉 = 0






latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708091-002-0015-001-0005000050010015002



latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708091-002-0015-001-0005000050010015002




latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708091-0025-002-0015-001-00050000500100150020025




int

d2x(

2partφpartφ +1

2|W prime|2 + ψMψ

)


ψ =

(

ψ1

ψ2






2W primeε2





1

2a

(






1

a2

(

sum

micro=12




2

sum


sum

x

(


1

2|Wx|2

)







)

=

(

Wxy 2partxy

2partxy Wxy

)

Wxy =partWx

partφy (57)28


sum

xy




2εξx δ(1)ψ1

x = 0









ar


S(1)bos =

a2

2

sum

x

(


+ a2sum

x

(




2(amr) a2

sum


(1)bos =

a2

2

sum

xy


pmicro =1


2

asin

(apmicro2





2arp2 plusmn i|





iar



iar


S(2)bos =

a2

2

sum

x

(


+ a2sum

x

(



S(2)bos =

a2

2

sum

xy


microp = m2 +

p2 +(

12ar p2


radic

p2 +(

12arp2




r2 =4

3(75)31





(3)bos =

a2

2

sum

xy


ϕ =

(

ϕ1

ϕ2

) and ξ =

(

ξ1

ξ2

)


2mφ2 + 1


W prime(φ) = mϕ1 + u+ i(mϕ2 + v) (80)32



(n)0 +m)ϕ +

(

u

v


S(n)bos =

a2

2

sum

xy

(ϕx K(n)xy ϕy) +

a2

2

sum

x


sum

x

ux

(

(div ϕ)x minusar

2(∆ϕ1)x

)

minus a2sum

x

vx

(

(13url ϕ)x +ar

2(∆ϕ2)x

)

∆(2) = a2sum

x

ux

(

(div ϕ)x minusar

2(∆ϕ2)x

)

+ a2sum

x

vx

(


2(∆ϕ1)x

)

(84)∆(3) = a2

sum

x


x



1

2mφ2 +

1


m1minusloop = m(

1 minus g2

4πm2






m


002 004 006 008 01 012 01466577588599510105


35


partH


partφx (87)where

H =1

2

sum

x




prod

x

(

1 +m+ 3gφ2x

)


partM [φ]xypartφz




37


g=2g=1

g=05g=02g=01

R

p(R

)

50-5-10-15-20-25

25

2

15

1

05


quencheddynamical

t

Gbos(t

)

35302520151050

0045

004

0035

003

0025

002

0015

001

0005


39




(

G(x)

G(x+ a)

)

G = G(n)bos or G


G(x) =sum



Gferm(x) =sum

λ


λ (94)40




Gferm(x) =1

N

sum

k

e2πiNkx

2πiNk +m


latti13e point x

〈ψxψ

0〉


0 10 20 30 40 50 60-010010203040506070809


1e-09

1e-08

1e-07

1e-06

1e-05

1e-04

0001

001

01

1

0 20 40 60 80 100 120


latti13e point x

|〈ψxψ

0〉|



1

NtN2x

sum

tprime

sum

xxprime


42

andGferm(t) =

1

NtN2x

2sum

α=1

sum

tprime

sum

xxprime



1









1


1



Iε + Iπ =

int

BZ

d2k

(2π)2

1


Iε =

int

D

d2k

(2π)2

1

(k2 +m2)((k + q1)2 +m2) ((k + qnminus1)2 +m2)

Iπ =

int

BZD

d2k

(2π)2

1


le (Ca2)nminus1

int

|k|leradic

2πa

d2k

(2π)2

1


(

1 +2π2

a2m2

)



int

BZ

d2k

(2π)2



I primeε =

int

D

d2k

(2π)2


I primeπ =

int

BZD

d2k

(2π)2

Pmicro(k) P(k + ql)


leint

BZD

d2k

(2π)2


le(π

a

)l+1

(Ca2)nminus1

int

|k|leradic

2πa

d2k

(2π)2

1


(

1 +2π2

a2m2

)



iar

2γ3∆ with r2 =

4



47

internal linesq

k

propint

BZ

d2k

(2π)2



int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)+

int

BZ

d2k

(2π)2

m2

(P 2(k) +m2)(P 2(k minus q) +m2)


=

int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)

=

int

BZ

d2k

(2π)2


(k2 +m2)(P (k minus q)2 +m2)

minus 2Λsum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2

P micro(k minus q)



sum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2


le 2Cπ2

int minusΛ+q1

minusΛ

dk1

2π

int Λ

minusΛ

dk2

2π

1


=C

2

∣

∣

∣

∣

int Λ

minusΛ


)

ω(k2)minus1

∣

∣

∣

∣


le C

2

int Λ

minusΛ

dk2

∣

∣

∣

q1m2 + k2

2 minus Λq1 + Λ2

∣

∣







52

Introduction


Lattice models












Ward identities









Algorithmic aspects

Quantum Mechanics




Summary and outlook











m


improved SLAC ferm


00102030405060708091

0 0005 001 0015 002 0025 003 0035

00102030405060708091121251313514145151551616517


int

D(φ ψ) eminusS+P


sum

y


(41)20


latti13e point x

R(1

)x

R(2)

x


0 5 10 15 20-0002-0001000010002




latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708090001-0002-0001000010002



latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708090001-0006-0004-00020000200040006







latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708090001-0006-0004-00020000200040006


2)〉 = 0






latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708091-002-0015-001-0005000050010015002



latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708091-002-0015-001-0005000050010015002




latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708091-0025-002-0015-001-00050000500100150020025




int

d2x(

2partφpartφ +1

2|W prime|2 + ψMψ

)


ψ =

(

ψ1

ψ2






2W primeε2





1

2a

(






1

a2

(

sum

micro=12




2

sum


sum

x

(


1

2|Wx|2

)







)

=

(

Wxy 2partxy

2partxy Wxy

)

Wxy =partWx

partφy (57)28


sum

xy




2εξx δ(1)ψ1

x = 0









ar


S(1)bos =

a2

2

sum

x

(


+ a2sum

x

(




2(amr) a2

sum


(1)bos =

a2

2

sum

xy


pmicro =1


2

asin

(apmicro2





2arp2 plusmn i|





iar



iar


S(2)bos =

a2

2

sum

x

(


+ a2sum

x

(



S(2)bos =

a2

2

sum

xy


microp = m2 +

p2 +(

12ar p2


radic

p2 +(

12arp2




r2 =4

3(75)31





(3)bos =

a2

2

sum

xy


ϕ =

(

ϕ1

ϕ2

) and ξ =

(

ξ1

ξ2

)


2mφ2 + 1


W prime(φ) = mϕ1 + u+ i(mϕ2 + v) (80)32



(n)0 +m)ϕ +

(

u

v


S(n)bos =

a2

2

sum

xy

(ϕx K(n)xy ϕy) +

a2

2

sum

x


sum

x

ux

(

(div ϕ)x minusar

2(∆ϕ1)x

)

minus a2sum

x

vx

(

(13url ϕ)x +ar

2(∆ϕ2)x

)

∆(2) = a2sum

x

ux

(

(div ϕ)x minusar

2(∆ϕ2)x

)

+ a2sum

x

vx

(


2(∆ϕ1)x

)

(84)∆(3) = a2

sum

x


x



1

2mφ2 +

1


m1minusloop = m(

1 minus g2

4πm2






m


002 004 006 008 01 012 01466577588599510105


35


partH


partφx (87)where

H =1

2

sum

x




prod

x

(

1 +m+ 3gφ2x

)


partM [φ]xypartφz




37


g=2g=1

g=05g=02g=01

R

p(R

)

50-5-10-15-20-25

25

2

15

1

05


quencheddynamical

t

Gbos(t

)

35302520151050

0045

004

0035

003

0025

002

0015

001

0005


39




(

G(x)

G(x+ a)

)

G = G(n)bos or G


G(x) =sum



Gferm(x) =sum

λ


λ (94)40




Gferm(x) =1

N

sum

k

e2πiNkx

2πiNk +m


latti13e point x

〈ψxψ

0〉


0 10 20 30 40 50 60-010010203040506070809


1e-09

1e-08

1e-07

1e-06

1e-05

1e-04

0001

001

01

1

0 20 40 60 80 100 120


latti13e point x

|〈ψxψ

0〉|



1

NtN2x

sum

tprime

sum

xxprime


42

andGferm(t) =

1

NtN2x

2sum

α=1

sum

tprime

sum

xxprime



1









1


1



Iε + Iπ =

int

BZ

d2k

(2π)2

1


Iε =

int

D

d2k

(2π)2

1

(k2 +m2)((k + q1)2 +m2) ((k + qnminus1)2 +m2)

Iπ =

int

BZD

d2k

(2π)2

1


le (Ca2)nminus1

int

|k|leradic

2πa

d2k

(2π)2

1


(

1 +2π2

a2m2

)



int

BZ

d2k

(2π)2



I primeε =

int

D

d2k

(2π)2


I primeπ =

int

BZD

d2k

(2π)2

Pmicro(k) P(k + ql)


leint

BZD

d2k

(2π)2


le(π

a

)l+1

(Ca2)nminus1

int

|k|leradic

2πa

d2k

(2π)2

1


(

1 +2π2

a2m2

)



iar

2γ3∆ with r2 =

4



47

internal linesq

k

propint

BZ

d2k

(2π)2



int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)+

int

BZ

d2k

(2π)2

m2

(P 2(k) +m2)(P 2(k minus q) +m2)


=

int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)

=

int

BZ

d2k

(2π)2


(k2 +m2)(P (k minus q)2 +m2)

minus 2Λsum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2

P micro(k minus q)



sum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2


le 2Cπ2

int minusΛ+q1

minusΛ

dk1

2π

int Λ

minusΛ

dk2

2π

1


=C

2

∣

∣

∣

∣

int Λ

minusΛ


)

ω(k2)minus1

∣

∣

∣

∣


le C

2

int Λ

minusΛ

dk2

∣

∣

∣

q1m2 + k2

2 minus Λq1 + Λ2

∣

∣







52

Introduction


Lattice models












Ward identities









Algorithmic aspects

Quantum Mechanics




Summary and outlook



latti13e point x

R(1

)x

R(2)

x


0 5 10 15 20-0002-0001000010002




latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708090001-0002-0001000010002



latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708090001-0006-0004-00020000200040006







latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708090001-0006-0004-00020000200040006


2)〉 = 0






latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708091-002-0015-001-0005000050010015002



latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708091-002-0015-001-0005000050010015002




latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708091-0025-002-0015-001-00050000500100150020025




int

d2x(

2partφpartφ +1

2|W prime|2 + ψMψ

)


ψ =

(

ψ1

ψ2






2W primeε2





1

2a

(






1

a2

(

sum

micro=12




2

sum


sum

x

(


1

2|Wx|2

)







)

=

(

Wxy 2partxy

2partxy Wxy

)

Wxy =partWx

partφy (57)28


sum

xy




2εξx δ(1)ψ1

x = 0









ar


S(1)bos =

a2

2

sum

x

(


+ a2sum

x

(




2(amr) a2

sum


(1)bos =

a2

2

sum

xy


pmicro =1


2

asin

(apmicro2





2arp2 plusmn i|





iar



iar


S(2)bos =

a2

2

sum

x

(


+ a2sum

x

(



S(2)bos =

a2

2

sum

xy


microp = m2 +

p2 +(

12ar p2


radic

p2 +(

12arp2




r2 =4

3(75)31





(3)bos =

a2

2

sum

xy


ϕ =

(

ϕ1

ϕ2

) and ξ =

(

ξ1

ξ2

)


2mφ2 + 1


W prime(φ) = mϕ1 + u+ i(mϕ2 + v) (80)32



(n)0 +m)ϕ +

(

u

v


S(n)bos =

a2

2

sum

xy

(ϕx K(n)xy ϕy) +

a2

2

sum

x


sum

x

ux

(

(div ϕ)x minusar

2(∆ϕ1)x

)

minus a2sum

x

vx

(

(13url ϕ)x +ar

2(∆ϕ2)x

)

∆(2) = a2sum

x

ux

(

(div ϕ)x minusar

2(∆ϕ2)x

)

+ a2sum

x

vx

(


2(∆ϕ1)x

)

(84)∆(3) = a2

sum

x


x



1

2mφ2 +

1


m1minusloop = m(

1 minus g2

4πm2






m


002 004 006 008 01 012 01466577588599510105


35


partH


partφx (87)where

H =1

2

sum

x




prod

x

(

1 +m+ 3gφ2x

)


partM [φ]xypartφz




37


g=2g=1

g=05g=02g=01

R

p(R

)

50-5-10-15-20-25

25

2

15

1

05


quencheddynamical

t

Gbos(t

)

35302520151050

0045

004

0035

003

0025

002

0015

001

0005


39




(

G(x)

G(x+ a)

)

G = G(n)bos or G


G(x) =sum



Gferm(x) =sum

λ


λ (94)40




Gferm(x) =1

N

sum

k

e2πiNkx

2πiNk +m


latti13e point x

〈ψxψ

0〉


0 10 20 30 40 50 60-010010203040506070809


1e-09

1e-08

1e-07

1e-06

1e-05

1e-04

0001

001

01

1

0 20 40 60 80 100 120


latti13e point x

|〈ψxψ

0〉|



1

NtN2x

sum

tprime

sum

xxprime


42

andGferm(t) =

1

NtN2x

2sum

α=1

sum

tprime

sum

xxprime



1









1


1



Iε + Iπ =

int

BZ

d2k

(2π)2

1


Iε =

int

D

d2k

(2π)2

1

(k2 +m2)((k + q1)2 +m2) ((k + qnminus1)2 +m2)

Iπ =

int

BZD

d2k

(2π)2

1


le (Ca2)nminus1

int

|k|leradic

2πa

d2k

(2π)2

1


(

1 +2π2

a2m2

)



int

BZ

d2k

(2π)2



I primeε =

int

D

d2k

(2π)2


I primeπ =

int

BZD

d2k

(2π)2

Pmicro(k) P(k + ql)


leint

BZD

d2k

(2π)2


le(π

a

)l+1

(Ca2)nminus1

int

|k|leradic

2πa

d2k

(2π)2

1


(

1 +2π2

a2m2

)



iar

2γ3∆ with r2 =

4



47

internal linesq

k

propint

BZ

d2k

(2π)2



int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)+

int

BZ

d2k

(2π)2

m2

(P 2(k) +m2)(P 2(k minus q) +m2)


=

int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)

=

int

BZ

d2k

(2π)2


(k2 +m2)(P (k minus q)2 +m2)

minus 2Λsum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2

P micro(k minus q)



sum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2


le 2Cπ2

int minusΛ+q1

minusΛ

dk1

2π

int Λ

minusΛ

dk2

2π

1


=C

2

∣

∣

∣

∣

int Λ

minusΛ


)

ω(k2)minus1

∣

∣

∣

∣


le C

2

int Λ

minusΛ

dk2

∣

∣

∣

q1m2 + k2

2 minus Λq1 + Λ2

∣

∣







52

Introduction


Lattice models












Ward identities









Algorithmic aspects

Quantum Mechanics




Summary and outlook



latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708090001-0002-0001000010002



latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708090001-0006-0004-00020000200040006







latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708090001-0006-0004-00020000200040006


2)〉 = 0






latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708091-002-0015-001-0005000050010015002



latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708091-002-0015-001-0005000050010015002




latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708091-0025-002-0015-001-00050000500100150020025




int

d2x(

2partφpartφ +1

2|W prime|2 + ψMψ

)


ψ =

(

ψ1

ψ2






2W primeε2





1

2a

(






1

a2

(

sum

micro=12




2

sum


sum

x

(


1

2|Wx|2

)







)

=

(

Wxy 2partxy

2partxy Wxy

)

Wxy =partWx

partφy (57)28


sum

xy




2εξx δ(1)ψ1

x = 0









ar


S(1)bos =

a2

2

sum

x

(


+ a2sum

x

(




2(amr) a2

sum


(1)bos =

a2

2

sum

xy


pmicro =1


2

asin

(apmicro2





2arp2 plusmn i|





iar



iar


S(2)bos =

a2

2

sum

x

(


+ a2sum

x

(



S(2)bos =

a2

2

sum

xy


microp = m2 +

p2 +(

12ar p2


radic

p2 +(

12arp2




r2 =4

3(75)31





(3)bos =

a2

2

sum

xy


ϕ =

(

ϕ1

ϕ2

) and ξ =

(

ξ1

ξ2

)


2mφ2 + 1


W prime(φ) = mϕ1 + u+ i(mϕ2 + v) (80)32



(n)0 +m)ϕ +

(

u

v


S(n)bos =

a2

2

sum

xy

(ϕx K(n)xy ϕy) +

a2

2

sum

x


sum

x

ux

(

(div ϕ)x minusar

2(∆ϕ1)x

)

minus a2sum

x

vx

(

(13url ϕ)x +ar

2(∆ϕ2)x

)

∆(2) = a2sum

x

ux

(

(div ϕ)x minusar

2(∆ϕ2)x

)

+ a2sum

x

vx

(


2(∆ϕ1)x

)

(84)∆(3) = a2

sum

x


x



1

2mφ2 +

1


m1minusloop = m(

1 minus g2

4πm2






m


002 004 006 008 01 012 01466577588599510105


35


partH


partφx (87)where

H =1

2

sum

x




prod

x

(

1 +m+ 3gφ2x

)


partM [φ]xypartφz




37


g=2g=1

g=05g=02g=01

R

p(R

)

50-5-10-15-20-25

25

2

15

1

05


quencheddynamical

t

Gbos(t

)

35302520151050

0045

004

0035

003

0025

002

0015

001

0005


39




(

G(x)

G(x+ a)

)

G = G(n)bos or G


G(x) =sum



Gferm(x) =sum

λ


λ (94)40




Gferm(x) =1

N

sum

k

e2πiNkx

2πiNk +m


latti13e point x

〈ψxψ

0〉


0 10 20 30 40 50 60-010010203040506070809


1e-09

1e-08

1e-07

1e-06

1e-05

1e-04

0001

001

01

1

0 20 40 60 80 100 120


latti13e point x

|〈ψxψ

0〉|



1

NtN2x

sum

tprime

sum

xxprime


42

andGferm(t) =

1

NtN2x

2sum

α=1

sum

tprime

sum

xxprime



1









1


1



Iε + Iπ =

int

BZ

d2k

(2π)2

1


Iε =

int

D

d2k

(2π)2

1

(k2 +m2)((k + q1)2 +m2) ((k + qnminus1)2 +m2)

Iπ =

int

BZD

d2k

(2π)2

1


le (Ca2)nminus1

int

|k|leradic

2πa

d2k

(2π)2

1


(

1 +2π2

a2m2

)



int

BZ

d2k

(2π)2



I primeε =

int

D

d2k

(2π)2


I primeπ =

int

BZD

d2k

(2π)2

Pmicro(k) P(k + ql)


leint

BZD

d2k

(2π)2


le(π

a

)l+1

(Ca2)nminus1

int

|k|leradic

2πa

d2k

(2π)2

1


(

1 +2π2

a2m2

)



iar

2γ3∆ with r2 =

4



47

internal linesq

k

propint

BZ

d2k

(2π)2



int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)+

int

BZ

d2k

(2π)2

m2

(P 2(k) +m2)(P 2(k minus q) +m2)


=

int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)

=

int

BZ

d2k

(2π)2


(k2 +m2)(P (k minus q)2 +m2)

minus 2Λsum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2

P micro(k minus q)



sum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2


le 2Cπ2

int minusΛ+q1

minusΛ

dk1

2π

int Λ

minusΛ

dk2

2π

1


=C

2

∣

∣

∣

∣

int Λ

minusΛ


)

ω(k2)minus1

∣

∣

∣

∣


le C

2

int Λ

minusΛ

dk2

∣

∣

∣

q1m2 + k2

2 minus Λq1 + Λ2

∣

∣







52

Introduction


Lattice models












Ward identities









Algorithmic aspects

Quantum Mechanics




Summary and outlook



latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708090001-0006-0004-00020000200040006







latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708090001-0006-0004-00020000200040006


2)〉 = 0






latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708091-002-0015-001-0005000050010015002



latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708091-002-0015-001-0005000050010015002




latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708091-0025-002-0015-001-00050000500100150020025




int

d2x(

2partφpartφ +1

2|W prime|2 + ψMψ

)


ψ =

(

ψ1

ψ2






2W primeε2





1

2a

(






1

a2

(

sum

micro=12




2

sum


sum

x

(


1

2|Wx|2

)







)

=

(

Wxy 2partxy

2partxy Wxy

)

Wxy =partWx

partφy (57)28


sum

xy




2εξx δ(1)ψ1

x = 0









ar


S(1)bos =

a2

2

sum

x

(


+ a2sum

x

(




2(amr) a2

sum


(1)bos =

a2

2

sum

xy


pmicro =1


2

asin

(apmicro2





2arp2 plusmn i|





iar



iar


S(2)bos =

a2

2

sum

x

(


+ a2sum

x

(



S(2)bos =

a2

2

sum

xy


microp = m2 +

p2 +(

12ar p2


radic

p2 +(

12arp2




r2 =4

3(75)31





(3)bos =

a2

2

sum

xy


ϕ =

(

ϕ1

ϕ2

) and ξ =

(

ξ1

ξ2

)


2mφ2 + 1


W prime(φ) = mϕ1 + u+ i(mϕ2 + v) (80)32



(n)0 +m)ϕ +

(

u

v


S(n)bos =

a2

2

sum

xy

(ϕx K(n)xy ϕy) +

a2

2

sum

x


sum

x

ux

(

(div ϕ)x minusar

2(∆ϕ1)x

)

minus a2sum

x

vx

(

(13url ϕ)x +ar

2(∆ϕ2)x

)

∆(2) = a2sum

x

ux

(

(div ϕ)x minusar

2(∆ϕ2)x

)

+ a2sum

x

vx

(


2(∆ϕ1)x

)

(84)∆(3) = a2

sum

x


x



1

2mφ2 +

1


m1minusloop = m(

1 minus g2

4πm2






m


002 004 006 008 01 012 01466577588599510105


35


partH


partφx (87)where

H =1

2

sum

x




prod

x

(

1 +m+ 3gφ2x

)


partM [φ]xypartφz




37


g=2g=1

g=05g=02g=01

R

p(R

)

50-5-10-15-20-25

25

2

15

1

05


quencheddynamical

t

Gbos(t

)

35302520151050

0045

004

0035

003

0025

002

0015

001

0005


39




(

G(x)

G(x+ a)

)

G = G(n)bos or G


G(x) =sum



Gferm(x) =sum

λ


λ (94)40




Gferm(x) =1

N

sum

k

e2πiNkx

2πiNk +m


latti13e point x

〈ψxψ

0〉


0 10 20 30 40 50 60-010010203040506070809


1e-09

1e-08

1e-07

1e-06

1e-05

1e-04

0001

001

01

1

0 20 40 60 80 100 120


latti13e point x

|〈ψxψ

0〉|



1

NtN2x

sum

tprime

sum

xxprime


42

andGferm(t) =

1

NtN2x

2sum

α=1

sum

tprime

sum

xxprime



1









1


1



Iε + Iπ =

int

BZ

d2k

(2π)2

1


Iε =

int

D

d2k

(2π)2

1

(k2 +m2)((k + q1)2 +m2) ((k + qnminus1)2 +m2)

Iπ =

int

BZD

d2k

(2π)2

1


le (Ca2)nminus1

int

|k|leradic

2πa

d2k

(2π)2

1


(

1 +2π2

a2m2

)



int

BZ

d2k

(2π)2



I primeε =

int

D

d2k

(2π)2


I primeπ =

int

BZD

d2k

(2π)2

Pmicro(k) P(k + ql)


leint

BZD

d2k

(2π)2


le(π

a

)l+1

(Ca2)nminus1

int

|k|leradic

2πa

d2k

(2π)2

1


(

1 +2π2

a2m2

)



iar

2γ3∆ with r2 =

4



47

internal linesq

k

propint

BZ

d2k

(2π)2



int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)+

int

BZ

d2k

(2π)2

m2

(P 2(k) +m2)(P 2(k minus q) +m2)


=

int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)

=

int

BZ

d2k

(2π)2


(k2 +m2)(P (k minus q)2 +m2)

minus 2Λsum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2

P micro(k minus q)



sum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2


le 2Cπ2

int minusΛ+q1

minusΛ

dk1

2π

int Λ

minusΛ

dk2

2π

1


=C

2

∣

∣

∣

∣

int Λ

minusΛ


)

ω(k2)minus1

∣

∣

∣

∣


le C

2

int Λ

minusΛ

dk2

∣

∣

∣

q1m2 + k2

2 minus Λq1 + Λ2

∣

∣







52

Introduction


Lattice models












Ward identities









Algorithmic aspects

Quantum Mechanics




Summary and outlook




latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708090001-0006-0004-00020000200040006


2)〉 = 0






latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708091-002-0015-001-0005000050010015002



latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708091-002-0015-001-0005000050010015002




latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708091-0025-002-0015-001-00050000500100150020025




int

d2x(

2partφpartφ +1

2|W prime|2 + ψMψ

)


ψ =

(

ψ1

ψ2






2W primeε2





1

2a

(






1

a2

(

sum

micro=12




2

sum


sum

x

(


1

2|Wx|2

)







)

=

(

Wxy 2partxy

2partxy Wxy

)

Wxy =partWx

partφy (57)28


sum

xy




2εξx δ(1)ψ1

x = 0









ar


S(1)bos =

a2

2

sum

x

(


+ a2sum

x

(




2(amr) a2

sum


(1)bos =

a2

2

sum

xy


pmicro =1


2

asin

(apmicro2





2arp2 plusmn i|





iar



iar


S(2)bos =

a2

2

sum

x

(


+ a2sum

x

(



S(2)bos =

a2

2

sum

xy


microp = m2 +

p2 +(

12ar p2


radic

p2 +(

12arp2




r2 =4

3(75)31





(3)bos =

a2

2

sum

xy


ϕ =

(

ϕ1

ϕ2

) and ξ =

(

ξ1

ξ2

)


2mφ2 + 1


W prime(φ) = mϕ1 + u+ i(mϕ2 + v) (80)32



(n)0 +m)ϕ +

(

u

v


S(n)bos =

a2

2

sum

xy

(ϕx K(n)xy ϕy) +

a2

2

sum

x


sum

x

ux

(

(div ϕ)x minusar

2(∆ϕ1)x

)

minus a2sum

x

vx

(

(13url ϕ)x +ar

2(∆ϕ2)x

)

∆(2) = a2sum

x

ux

(

(div ϕ)x minusar

2(∆ϕ2)x

)

+ a2sum

x

vx

(


2(∆ϕ1)x

)

(84)∆(3) = a2

sum

x


x



1

2mφ2 +

1


m1minusloop = m(

1 minus g2

4πm2






m


002 004 006 008 01 012 01466577588599510105


35


partH


partφx (87)where

H =1

2

sum

x




prod

x

(

1 +m+ 3gφ2x

)


partM [φ]xypartφz




37


g=2g=1

g=05g=02g=01

R

p(R

)

50-5-10-15-20-25

25

2

15

1

05


quencheddynamical

t

Gbos(t

)

35302520151050

0045

004

0035

003

0025

002

0015

001

0005


39




(

G(x)

G(x+ a)

)

G = G(n)bos or G


G(x) =sum



Gferm(x) =sum

λ


λ (94)40




Gferm(x) =1

N

sum

k

e2πiNkx

2πiNk +m


latti13e point x

〈ψxψ

0〉


0 10 20 30 40 50 60-010010203040506070809


1e-09

1e-08

1e-07

1e-06

1e-05

1e-04

0001

001

01

1

0 20 40 60 80 100 120


latti13e point x

|〈ψxψ

0〉|



1

NtN2x

sum

tprime

sum

xxprime


42

andGferm(t) =

1

NtN2x

2sum

α=1

sum

tprime

sum

xxprime



1









1


1



Iε + Iπ =

int

BZ

d2k

(2π)2

1


Iε =

int

D

d2k

(2π)2

1

(k2 +m2)((k + q1)2 +m2) ((k + qnminus1)2 +m2)

Iπ =

int

BZD

d2k

(2π)2

1


le (Ca2)nminus1

int

|k|leradic

2πa

d2k

(2π)2

1


(

1 +2π2

a2m2

)



int

BZ

d2k

(2π)2



I primeε =

int

D

d2k

(2π)2


I primeπ =

int

BZD

d2k

(2π)2

Pmicro(k) P(k + ql)


leint

BZD

d2k

(2π)2


le(π

a

)l+1

(Ca2)nminus1

int

|k|leradic

2πa

d2k

(2π)2

1


(

1 +2π2

a2m2

)



iar

2γ3∆ with r2 =

4



47

internal linesq

k

propint

BZ

d2k

(2π)2



int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)+

int

BZ

d2k

(2π)2

m2

(P 2(k) +m2)(P 2(k minus q) +m2)


=

int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)

=

int

BZ

d2k

(2π)2


(k2 +m2)(P (k minus q)2 +m2)

minus 2Λsum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2

P micro(k minus q)



sum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2


le 2Cπ2

int minusΛ+q1

minusΛ

dk1

2π

int Λ

minusΛ

dk2

2π

1


=C

2

∣

∣

∣

∣

int Λ

minusΛ


)

ω(k2)minus1

∣

∣

∣

∣


le C

2

int Λ

minusΛ

dk2

∣

∣

∣

q1m2 + k2

2 minus Λq1 + Λ2

∣

∣







52

Introduction


Lattice models












Ward identities









Algorithmic aspects

Quantum Mechanics




Summary and outlook



latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708091-002-0015-001-0005000050010015002



latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708091-002-0015-001-0005000050010015002




latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708091-0025-002-0015-001-00050000500100150020025




int

d2x(

2partφpartφ +1

2|W prime|2 + ψMψ

)


ψ =

(

ψ1

ψ2






2W primeε2





1

2a

(






1

a2

(

sum

micro=12




2

sum


sum

x

(


1

2|Wx|2

)







)

=

(

Wxy 2partxy

2partxy Wxy

)

Wxy =partWx

partφy (57)28


sum

xy




2εξx δ(1)ψ1

x = 0









ar


S(1)bos =

a2

2

sum

x

(


+ a2sum

x

(




2(amr) a2

sum


(1)bos =

a2

2

sum

xy


pmicro =1


2

asin

(apmicro2





2arp2 plusmn i|





iar



iar


S(2)bos =

a2

2

sum

x

(


+ a2sum

x

(



S(2)bos =

a2

2

sum

xy


microp = m2 +

p2 +(

12ar p2


radic

p2 +(

12arp2




r2 =4

3(75)31





(3)bos =

a2

2

sum

xy


ϕ =

(

ϕ1

ϕ2

) and ξ =

(

ξ1

ξ2

)


2mφ2 + 1


W prime(φ) = mϕ1 + u+ i(mϕ2 + v) (80)32



(n)0 +m)ϕ +

(

u

v


S(n)bos =

a2

2

sum

xy

(ϕx K(n)xy ϕy) +

a2

2

sum

x


sum

x

ux

(

(div ϕ)x minusar

2(∆ϕ1)x

)

minus a2sum

x

vx

(

(13url ϕ)x +ar

2(∆ϕ2)x

)

∆(2) = a2sum

x

ux

(

(div ϕ)x minusar

2(∆ϕ2)x

)

+ a2sum

x

vx

(


2(∆ϕ1)x

)

(84)∆(3) = a2

sum

x


x



1

2mφ2 +

1


m1minusloop = m(

1 minus g2

4πm2






m


002 004 006 008 01 012 01466577588599510105


35


partH


partφx (87)where

H =1

2

sum

x




prod

x

(

1 +m+ 3gφ2x

)


partM [φ]xypartφz




37


g=2g=1

g=05g=02g=01

R

p(R

)

50-5-10-15-20-25

25

2

15

1

05


quencheddynamical

t

Gbos(t

)

35302520151050

0045

004

0035

003

0025

002

0015

001

0005


39




(

G(x)

G(x+ a)

)

G = G(n)bos or G


G(x) =sum



Gferm(x) =sum

λ


λ (94)40




Gferm(x) =1

N

sum

k

e2πiNkx

2πiNk +m


latti13e point x

〈ψxψ

0〉


0 10 20 30 40 50 60-010010203040506070809


1e-09

1e-08

1e-07

1e-06

1e-05

1e-04

0001

001

01

1

0 20 40 60 80 100 120


latti13e point x

|〈ψxψ

0〉|



1

NtN2x

sum

tprime

sum

xxprime


42

andGferm(t) =

1

NtN2x

2sum

α=1

sum

tprime

sum

xxprime



1









1


1



Iε + Iπ =

int

BZ

d2k

(2π)2

1


Iε =

int

D

d2k

(2π)2

1

(k2 +m2)((k + q1)2 +m2) ((k + qnminus1)2 +m2)

Iπ =

int

BZD

d2k

(2π)2

1


le (Ca2)nminus1

int

|k|leradic

2πa

d2k

(2π)2

1


(

1 +2π2

a2m2

)



int

BZ

d2k

(2π)2



I primeε =

int

D

d2k

(2π)2


I primeπ =

int

BZD

d2k

(2π)2

Pmicro(k) P(k + ql)


leint

BZD

d2k

(2π)2


le(π

a

)l+1

(Ca2)nminus1

int

|k|leradic

2πa

d2k

(2π)2

1


(

1 +2π2

a2m2

)



iar

2γ3∆ with r2 =

4



47

internal linesq

k

propint

BZ

d2k

(2π)2



int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)+

int

BZ

d2k

(2π)2

m2

(P 2(k) +m2)(P 2(k minus q) +m2)


=

int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)

=

int

BZ

d2k

(2π)2


(k2 +m2)(P (k minus q)2 +m2)

minus 2Λsum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2

P micro(k minus q)



sum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2


le 2Cπ2

int minusΛ+q1

minusΛ

dk1

2π

int Λ

minusΛ

dk2

2π

1


=C

2

∣

∣

∣

∣

int Λ

minusΛ


)

ω(k2)minus1

∣

∣

∣

∣


le C

2

int Λ

minusΛ

dk2

∣

∣

∣

q1m2 + k2

2 minus Λq1 + Λ2

∣

∣







52

Introduction


Lattice models












Ward identities









Algorithmic aspects

Quantum Mechanics




Summary and outlook




latti13e point x

R(1

)x

R(2)

x

Ward-Identity 2


00102030405060708091

0 5 10 15 20

00102030405060708091-0025-002-0015-001-00050000500100150020025




int

d2x(

2partφpartφ +1

2|W prime|2 + ψMψ

)


ψ =

(

ψ1

ψ2






2W primeε2





1

2a

(






1

a2

(

sum

micro=12




2

sum


sum

x

(


1

2|Wx|2

)







)

=

(

Wxy 2partxy

2partxy Wxy

)

Wxy =partWx

partφy (57)28


sum

xy




2εξx δ(1)ψ1

x = 0









ar


S(1)bos =

a2

2

sum

x

(


+ a2sum

x

(




2(amr) a2

sum


(1)bos =

a2

2

sum

xy


pmicro =1


2

asin

(apmicro2





2arp2 plusmn i|





iar



iar


S(2)bos =

a2

2

sum

x

(


+ a2sum

x

(



S(2)bos =

a2

2

sum

xy


microp = m2 +

p2 +(

12ar p2


radic

p2 +(

12arp2




r2 =4

3(75)31





(3)bos =

a2

2

sum

xy


ϕ =

(

ϕ1

ϕ2

) and ξ =

(

ξ1

ξ2

)


2mφ2 + 1


W prime(φ) = mϕ1 + u+ i(mϕ2 + v) (80)32



(n)0 +m)ϕ +

(

u

v


S(n)bos =

a2

2

sum

xy

(ϕx K(n)xy ϕy) +

a2

2

sum

x


sum

x

ux

(

(div ϕ)x minusar

2(∆ϕ1)x

)

minus a2sum

x

vx

(

(13url ϕ)x +ar

2(∆ϕ2)x

)

∆(2) = a2sum

x

ux

(

(div ϕ)x minusar

2(∆ϕ2)x

)

+ a2sum

x

vx

(


2(∆ϕ1)x

)

(84)∆(3) = a2

sum

x


x



1

2mφ2 +

1


m1minusloop = m(

1 minus g2

4πm2






m


002 004 006 008 01 012 01466577588599510105


35


partH


partφx (87)where

H =1

2

sum

x




prod

x

(

1 +m+ 3gφ2x

)


partM [φ]xypartφz




37


g=2g=1

g=05g=02g=01

R

p(R

)

50-5-10-15-20-25

25

2

15

1

05


quencheddynamical

t

Gbos(t

)

35302520151050

0045

004

0035

003

0025

002

0015

001

0005


39




(

G(x)

G(x+ a)

)

G = G(n)bos or G


G(x) =sum



Gferm(x) =sum

λ


λ (94)40




Gferm(x) =1

N

sum

k

e2πiNkx

2πiNk +m


latti13e point x

〈ψxψ

0〉


0 10 20 30 40 50 60-010010203040506070809


1e-09

1e-08

1e-07

1e-06

1e-05

1e-04

0001

001

01

1

0 20 40 60 80 100 120


latti13e point x

|〈ψxψ

0〉|



1

NtN2x

sum

tprime

sum

xxprime


42

andGferm(t) =

1

NtN2x

2sum

α=1

sum

tprime

sum

xxprime



1









1


1



Iε + Iπ =

int

BZ

d2k

(2π)2

1


Iε =

int

D

d2k

(2π)2

1

(k2 +m2)((k + q1)2 +m2) ((k + qnminus1)2 +m2)

Iπ =

int

BZD

d2k

(2π)2

1


le (Ca2)nminus1

int

|k|leradic

2πa

d2k

(2π)2

1


(

1 +2π2

a2m2

)



int

BZ

d2k

(2π)2



I primeε =

int

D

d2k

(2π)2


I primeπ =

int

BZD

d2k

(2π)2

Pmicro(k) P(k + ql)


leint

BZD

d2k

(2π)2


le(π

a

)l+1

(Ca2)nminus1

int

|k|leradic

2πa

d2k

(2π)2

1


(

1 +2π2

a2m2

)



iar

2γ3∆ with r2 =

4



47

internal linesq

k

propint

BZ

d2k

(2π)2



int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)+

int

BZ

d2k

(2π)2

m2

(P 2(k) +m2)(P 2(k minus q) +m2)


=

int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)

=

int

BZ

d2k

(2π)2


(k2 +m2)(P (k minus q)2 +m2)

minus 2Λsum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2

P micro(k minus q)



sum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2


le 2Cπ2

int minusΛ+q1

minusΛ

dk1

2π

int Λ

minusΛ

dk2

2π

1


=C

2

∣

∣

∣

∣

int Λ

minusΛ


)

ω(k2)minus1

∣

∣

∣

∣


le C

2

int Λ

minusΛ

dk2

∣

∣

∣

q1m2 + k2

2 minus Λq1 + Λ2

∣

∣







52

Introduction


Lattice models












Ward identities









Algorithmic aspects

Quantum Mechanics




Summary and outlook




int

d2x(

2partφpartφ +1

2|W prime|2 + ψMψ

)


ψ =

(

ψ1

ψ2






2W primeε2





1

2a

(






1

a2

(

sum

micro=12




2

sum


sum

x

(


1

2|Wx|2

)







)

=

(

Wxy 2partxy

2partxy Wxy

)

Wxy =partWx

partφy (57)28


sum

xy




2εξx δ(1)ψ1

x = 0









ar


S(1)bos =

a2

2

sum

x

(


+ a2sum

x

(




2(amr) a2

sum


(1)bos =

a2

2

sum

xy


pmicro =1


2

asin

(apmicro2





2arp2 plusmn i|





iar



iar


S(2)bos =

a2

2

sum

x

(


+ a2sum

x

(



S(2)bos =

a2

2

sum

xy


microp = m2 +

p2 +(

12ar p2


radic

p2 +(

12arp2




r2 =4

3(75)31





(3)bos =

a2

2

sum

xy


ϕ =

(

ϕ1

ϕ2

) and ξ =

(

ξ1

ξ2

)


2mφ2 + 1


W prime(φ) = mϕ1 + u+ i(mϕ2 + v) (80)32



(n)0 +m)ϕ +

(

u

v


S(n)bos =

a2

2

sum

xy

(ϕx K(n)xy ϕy) +

a2

2

sum

x


sum

x

ux

(

(div ϕ)x minusar

2(∆ϕ1)x

)

minus a2sum

x

vx

(

(13url ϕ)x +ar

2(∆ϕ2)x

)

∆(2) = a2sum

x

ux

(

(div ϕ)x minusar

2(∆ϕ2)x

)

+ a2sum

x

vx

(


2(∆ϕ1)x

)

(84)∆(3) = a2

sum

x


x



1

2mφ2 +

1


m1minusloop = m(

1 minus g2

4πm2






m


002 004 006 008 01 012 01466577588599510105


35


partH


partφx (87)where

H =1

2

sum

x




prod

x

(

1 +m+ 3gφ2x

)


partM [φ]xypartφz




37


g=2g=1

g=05g=02g=01

R

p(R

)

50-5-10-15-20-25

25

2

15

1

05


quencheddynamical

t

Gbos(t

)

35302520151050

0045

004

0035

003

0025

002

0015

001

0005


39




(

G(x)

G(x+ a)

)

G = G(n)bos or G


G(x) =sum



Gferm(x) =sum

λ


λ (94)40




Gferm(x) =1

N

sum

k

e2πiNkx

2πiNk +m


latti13e point x

〈ψxψ

0〉


0 10 20 30 40 50 60-010010203040506070809


1e-09

1e-08

1e-07

1e-06

1e-05

1e-04

0001

001

01

1

0 20 40 60 80 100 120


latti13e point x

|〈ψxψ

0〉|



1

NtN2x

sum

tprime

sum

xxprime


42

andGferm(t) =

1

NtN2x

2sum

α=1

sum

tprime

sum

xxprime



1









1


1



Iε + Iπ =

int

BZ

d2k

(2π)2

1


Iε =

int

D

d2k

(2π)2

1

(k2 +m2)((k + q1)2 +m2) ((k + qnminus1)2 +m2)

Iπ =

int

BZD

d2k

(2π)2

1


le (Ca2)nminus1

int

|k|leradic

2πa

d2k

(2π)2

1


(

1 +2π2

a2m2

)



int

BZ

d2k

(2π)2



I primeε =

int

D

d2k

(2π)2


I primeπ =

int

BZD

d2k

(2π)2

Pmicro(k) P(k + ql)


leint

BZD

d2k

(2π)2


le(π

a

)l+1

(Ca2)nminus1

int

|k|leradic

2πa

d2k

(2π)2

1


(

1 +2π2

a2m2

)



iar

2γ3∆ with r2 =

4



47

internal linesq

k

propint

BZ

d2k

(2π)2



int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)+

int

BZ

d2k

(2π)2

m2

(P 2(k) +m2)(P 2(k minus q) +m2)


=

int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)

=

int

BZ

d2k

(2π)2


(k2 +m2)(P (k minus q)2 +m2)

minus 2Λsum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2

P micro(k minus q)



sum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2


le 2Cπ2

int minusΛ+q1

minusΛ

dk1

2π

int Λ

minusΛ

dk2

2π

1


=C

2

∣

∣

∣

∣

int Λ

minusΛ


)

ω(k2)minus1

∣

∣

∣

∣


le C

2

int Λ

minusΛ

dk2

∣

∣

∣

q1m2 + k2

2 minus Λq1 + Λ2

∣

∣







52

Introduction


Lattice models












Ward identities









Algorithmic aspects

Quantum Mechanics




Summary and outlook



1

2a

(






1

a2

(

sum

micro=12




2

sum


sum

x

(


1

2|Wx|2

)







)

=

(

Wxy 2partxy

2partxy Wxy

)

Wxy =partWx

partφy (57)28


sum

xy




2εξx δ(1)ψ1

x = 0









ar


S(1)bos =

a2

2

sum

x

(


+ a2sum

x

(




2(amr) a2

sum


(1)bos =

a2

2

sum

xy


pmicro =1


2

asin

(apmicro2





2arp2 plusmn i|





iar



iar


S(2)bos =

a2

2

sum

x

(


+ a2sum

x

(



S(2)bos =

a2

2

sum

xy


microp = m2 +

p2 +(

12ar p2


radic

p2 +(

12arp2




r2 =4

3(75)31





(3)bos =

a2

2

sum

xy


ϕ =

(

ϕ1

ϕ2

) and ξ =

(

ξ1

ξ2

)


2mφ2 + 1


W prime(φ) = mϕ1 + u+ i(mϕ2 + v) (80)32



(n)0 +m)ϕ +

(

u

v


S(n)bos =

a2

2

sum

xy

(ϕx K(n)xy ϕy) +

a2

2

sum

x


sum

x

ux

(

(div ϕ)x minusar

2(∆ϕ1)x

)

minus a2sum

x

vx

(

(13url ϕ)x +ar

2(∆ϕ2)x

)

∆(2) = a2sum

x

ux

(

(div ϕ)x minusar

2(∆ϕ2)x

)

+ a2sum

x

vx

(


2(∆ϕ1)x

)

(84)∆(3) = a2

sum

x


x



1

2mφ2 +

1


m1minusloop = m(

1 minus g2

4πm2






m


002 004 006 008 01 012 01466577588599510105


35


partH


partφx (87)where

H =1

2

sum

x




prod

x

(

1 +m+ 3gφ2x

)


partM [φ]xypartφz




37


g=2g=1

g=05g=02g=01

R

p(R

)

50-5-10-15-20-25

25

2

15

1

05


quencheddynamical

t

Gbos(t

)

35302520151050

0045

004

0035

003

0025

002

0015

001

0005


39




(

G(x)

G(x+ a)

)

G = G(n)bos or G


G(x) =sum



Gferm(x) =sum

λ


λ (94)40




Gferm(x) =1

N

sum

k

e2πiNkx

2πiNk +m


latti13e point x

〈ψxψ

0〉


0 10 20 30 40 50 60-010010203040506070809


1e-09

1e-08

1e-07

1e-06

1e-05

1e-04

0001

001

01

1

0 20 40 60 80 100 120


latti13e point x

|〈ψxψ

0〉|



1

NtN2x

sum

tprime

sum

xxprime


42

andGferm(t) =

1

NtN2x

2sum

α=1

sum

tprime

sum

xxprime



1









1


1



Iε + Iπ =

int

BZ

d2k

(2π)2

1


Iε =

int

D

d2k

(2π)2

1

(k2 +m2)((k + q1)2 +m2) ((k + qnminus1)2 +m2)

Iπ =

int

BZD

d2k

(2π)2

1


le (Ca2)nminus1

int

|k|leradic

2πa

d2k

(2π)2

1


(

1 +2π2

a2m2

)



int

BZ

d2k

(2π)2



I primeε =

int

D

d2k

(2π)2


I primeπ =

int

BZD

d2k

(2π)2

Pmicro(k) P(k + ql)


leint

BZD

d2k

(2π)2


le(π

a

)l+1

(Ca2)nminus1

int

|k|leradic

2πa

d2k

(2π)2

1


(

1 +2π2

a2m2

)



iar

2γ3∆ with r2 =

4



47

internal linesq

k

propint

BZ

d2k

(2π)2



int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)+

int

BZ

d2k

(2π)2

m2

(P 2(k) +m2)(P 2(k minus q) +m2)


=

int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)

=

int

BZ

d2k

(2π)2


(k2 +m2)(P (k minus q)2 +m2)

minus 2Λsum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2

P micro(k minus q)



sum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2


le 2Cπ2

int minusΛ+q1

minusΛ

dk1

2π

int Λ

minusΛ

dk2

2π

1


=C

2

∣

∣

∣

∣

int Λ

minusΛ


)

ω(k2)minus1

∣

∣

∣

∣


le C

2

int Λ

minusΛ

dk2

∣

∣

∣

q1m2 + k2

2 minus Λq1 + Λ2

∣

∣







52

Introduction


Lattice models












Ward identities









Algorithmic aspects

Quantum Mechanics




Summary and outlook



sum

xy




2εξx δ(1)ψ1

x = 0









ar


S(1)bos =

a2

2

sum

x

(


+ a2sum

x

(




2(amr) a2

sum


(1)bos =

a2

2

sum

xy


pmicro =1


2

asin

(apmicro2





2arp2 plusmn i|





iar



iar


S(2)bos =

a2

2

sum

x

(


+ a2sum

x

(



S(2)bos =

a2

2

sum

xy


microp = m2 +

p2 +(

12ar p2


radic

p2 +(

12arp2




r2 =4

3(75)31





(3)bos =

a2

2

sum

xy


ϕ =

(

ϕ1

ϕ2

) and ξ =

(

ξ1

ξ2

)


2mφ2 + 1


W prime(φ) = mϕ1 + u+ i(mϕ2 + v) (80)32



(n)0 +m)ϕ +

(

u

v


S(n)bos =

a2

2

sum

xy

(ϕx K(n)xy ϕy) +

a2

2

sum

x


sum

x

ux

(

(div ϕ)x minusar

2(∆ϕ1)x

)

minus a2sum

x

vx

(

(13url ϕ)x +ar

2(∆ϕ2)x

)

∆(2) = a2sum

x

ux

(

(div ϕ)x minusar

2(∆ϕ2)x

)

+ a2sum

x

vx

(


2(∆ϕ1)x

)

(84)∆(3) = a2

sum

x


x



1

2mφ2 +

1


m1minusloop = m(

1 minus g2

4πm2






m


002 004 006 008 01 012 01466577588599510105


35


partH


partφx (87)where

H =1

2

sum

x




prod

x

(

1 +m+ 3gφ2x

)


partM [φ]xypartφz




37


g=2g=1

g=05g=02g=01

R

p(R

)

50-5-10-15-20-25

25

2

15

1

05


quencheddynamical

t

Gbos(t

)

35302520151050

0045

004

0035

003

0025

002

0015

001

0005


39




(

G(x)

G(x+ a)

)

G = G(n)bos or G


G(x) =sum



Gferm(x) =sum

λ


λ (94)40




Gferm(x) =1

N

sum

k

e2πiNkx

2πiNk +m


latti13e point x

〈ψxψ

0〉


0 10 20 30 40 50 60-010010203040506070809


1e-09

1e-08

1e-07

1e-06

1e-05

1e-04

0001

001

01

1

0 20 40 60 80 100 120


latti13e point x

|〈ψxψ

0〉|



1

NtN2x

sum

tprime

sum

xxprime


42

andGferm(t) =

1

NtN2x

2sum

α=1

sum

tprime

sum

xxprime



1









1


1



Iε + Iπ =

int

BZ

d2k

(2π)2

1


Iε =

int

D

d2k

(2π)2

1

(k2 +m2)((k + q1)2 +m2) ((k + qnminus1)2 +m2)

Iπ =

int

BZD

d2k

(2π)2

1


le (Ca2)nminus1

int

|k|leradic

2πa

d2k

(2π)2

1


(

1 +2π2

a2m2

)



int

BZ

d2k

(2π)2



I primeε =

int

D

d2k

(2π)2


I primeπ =

int

BZD

d2k

(2π)2

Pmicro(k) P(k + ql)


leint

BZD

d2k

(2π)2


le(π

a

)l+1

(Ca2)nminus1

int

|k|leradic

2πa

d2k

(2π)2

1


(

1 +2π2

a2m2

)



iar

2γ3∆ with r2 =

4



47

internal linesq

k

propint

BZ

d2k

(2π)2



int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)+

int

BZ

d2k

(2π)2

m2

(P 2(k) +m2)(P 2(k minus q) +m2)


=

int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)

=

int

BZ

d2k

(2π)2


(k2 +m2)(P (k minus q)2 +m2)

minus 2Λsum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2

P micro(k minus q)



sum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2


le 2Cπ2

int minusΛ+q1

minusΛ

dk1

2π

int Λ

minusΛ

dk2

2π

1


=C

2

∣

∣

∣

∣

int Λ

minusΛ


)

ω(k2)minus1

∣

∣

∣

∣


le C

2

int Λ

minusΛ

dk2

∣

∣

∣

q1m2 + k2

2 minus Λq1 + Λ2

∣

∣







52

Introduction


Lattice models












Ward identities









Algorithmic aspects

Quantum Mechanics




Summary and outlook




ar


S(1)bos =

a2

2

sum

x

(


+ a2sum

x

(




2(amr) a2

sum


(1)bos =

a2

2

sum

xy


pmicro =1


2

asin

(apmicro2





2arp2 plusmn i|





iar



iar


S(2)bos =

a2

2

sum

x

(


+ a2sum

x

(



S(2)bos =

a2

2

sum

xy


microp = m2 +

p2 +(

12ar p2


radic

p2 +(

12arp2




r2 =4

3(75)31





(3)bos =

a2

2

sum

xy


ϕ =

(

ϕ1

ϕ2

) and ξ =

(

ξ1

ξ2

)


2mφ2 + 1


W prime(φ) = mϕ1 + u+ i(mϕ2 + v) (80)32



(n)0 +m)ϕ +

(

u

v


S(n)bos =

a2

2

sum

xy

(ϕx K(n)xy ϕy) +

a2

2

sum

x


sum

x

ux

(

(div ϕ)x minusar

2(∆ϕ1)x

)

minus a2sum

x

vx

(

(13url ϕ)x +ar

2(∆ϕ2)x

)

∆(2) = a2sum

x

ux

(

(div ϕ)x minusar

2(∆ϕ2)x

)

+ a2sum

x

vx

(


2(∆ϕ1)x

)

(84)∆(3) = a2

sum

x


x



1

2mφ2 +

1


m1minusloop = m(

1 minus g2

4πm2






m


002 004 006 008 01 012 01466577588599510105


35


partH


partφx (87)where

H =1

2

sum

x




prod

x

(

1 +m+ 3gφ2x

)


partM [φ]xypartφz




37


g=2g=1

g=05g=02g=01

R

p(R

)

50-5-10-15-20-25

25

2

15

1

05


quencheddynamical

t

Gbos(t

)

35302520151050

0045

004

0035

003

0025

002

0015

001

0005


39




(

G(x)

G(x+ a)

)

G = G(n)bos or G


G(x) =sum



Gferm(x) =sum

λ


λ (94)40




Gferm(x) =1

N

sum

k

e2πiNkx

2πiNk +m


latti13e point x

〈ψxψ

0〉


0 10 20 30 40 50 60-010010203040506070809


1e-09

1e-08

1e-07

1e-06

1e-05

1e-04

0001

001

01

1

0 20 40 60 80 100 120


latti13e point x

|〈ψxψ

0〉|



1

NtN2x

sum

tprime

sum

xxprime


42

andGferm(t) =

1

NtN2x

2sum

α=1

sum

tprime

sum

xxprime



1









1


1



Iε + Iπ =

int

BZ

d2k

(2π)2

1


Iε =

int

D

d2k

(2π)2

1

(k2 +m2)((k + q1)2 +m2) ((k + qnminus1)2 +m2)

Iπ =

int

BZD

d2k

(2π)2

1


le (Ca2)nminus1

int

|k|leradic

2πa

d2k

(2π)2

1


(

1 +2π2

a2m2

)



int

BZ

d2k

(2π)2



I primeε =

int

D

d2k

(2π)2


I primeπ =

int

BZD

d2k

(2π)2

Pmicro(k) P(k + ql)


leint

BZD

d2k

(2π)2


le(π

a

)l+1

(Ca2)nminus1

int

|k|leradic

2πa

d2k

(2π)2

1


(

1 +2π2

a2m2

)



iar

2γ3∆ with r2 =

4



47

internal linesq

k

propint

BZ

d2k

(2π)2



int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)+

int

BZ

d2k

(2π)2

m2

(P 2(k) +m2)(P 2(k minus q) +m2)


=

int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)

=

int

BZ

d2k

(2π)2


(k2 +m2)(P (k minus q)2 +m2)

minus 2Λsum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2

P micro(k minus q)



sum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2


le 2Cπ2

int minusΛ+q1

minusΛ

dk1

2π

int Λ

minusΛ

dk2

2π

1


=C

2

∣

∣

∣

∣

int Λ

minusΛ


)

ω(k2)minus1

∣

∣

∣

∣


le C

2

int Λ

minusΛ

dk2

∣

∣

∣

q1m2 + k2

2 minus Λq1 + Λ2

∣

∣







52

Introduction


Lattice models












Ward identities









Algorithmic aspects

Quantum Mechanics




Summary and outlook



iar



iar


S(2)bos =

a2

2

sum

x

(


+ a2sum

x

(



S(2)bos =

a2

2

sum

xy


microp = m2 +

p2 +(

12ar p2


radic

p2 +(

12arp2




r2 =4

3(75)31





(3)bos =

a2

2

sum

xy


ϕ =

(

ϕ1

ϕ2

) and ξ =

(

ξ1

ξ2

)


2mφ2 + 1


W prime(φ) = mϕ1 + u+ i(mϕ2 + v) (80)32



(n)0 +m)ϕ +

(

u

v


S(n)bos =

a2

2

sum

xy

(ϕx K(n)xy ϕy) +

a2

2

sum

x


sum

x

ux

(

(div ϕ)x minusar

2(∆ϕ1)x

)

minus a2sum

x

vx

(

(13url ϕ)x +ar

2(∆ϕ2)x

)

∆(2) = a2sum

x

ux

(

(div ϕ)x minusar

2(∆ϕ2)x

)

+ a2sum

x

vx

(


2(∆ϕ1)x

)

(84)∆(3) = a2

sum

x


x



1

2mφ2 +

1


m1minusloop = m(

1 minus g2

4πm2






m


002 004 006 008 01 012 01466577588599510105


35


partH


partφx (87)where

H =1

2

sum

x




prod

x

(

1 +m+ 3gφ2x

)


partM [φ]xypartφz




37


g=2g=1

g=05g=02g=01

R

p(R

)

50-5-10-15-20-25

25

2

15

1

05


quencheddynamical

t

Gbos(t

)

35302520151050

0045

004

0035

003

0025

002

0015

001

0005


39




(

G(x)

G(x+ a)

)

G = G(n)bos or G


G(x) =sum



Gferm(x) =sum

λ


λ (94)40




Gferm(x) =1

N

sum

k

e2πiNkx

2πiNk +m


latti13e point x

〈ψxψ

0〉


0 10 20 30 40 50 60-010010203040506070809


1e-09

1e-08

1e-07

1e-06

1e-05

1e-04

0001

001

01

1

0 20 40 60 80 100 120


latti13e point x

|〈ψxψ

0〉|



1

NtN2x

sum

tprime

sum

xxprime


42

andGferm(t) =

1

NtN2x

2sum

α=1

sum

tprime

sum

xxprime



1









1


1



Iε + Iπ =

int

BZ

d2k

(2π)2

1


Iε =

int

D

d2k

(2π)2

1

(k2 +m2)((k + q1)2 +m2) ((k + qnminus1)2 +m2)

Iπ =

int

BZD

d2k

(2π)2

1


le (Ca2)nminus1

int

|k|leradic

2πa

d2k

(2π)2

1


(

1 +2π2

a2m2

)



int

BZ

d2k

(2π)2



I primeε =

int

D

d2k

(2π)2


I primeπ =

int

BZD

d2k

(2π)2

Pmicro(k) P(k + ql)


leint

BZD

d2k

(2π)2


le(π

a

)l+1

(Ca2)nminus1

int

|k|leradic

2πa

d2k

(2π)2

1


(

1 +2π2

a2m2

)



iar

2γ3∆ with r2 =

4



47

internal linesq

k

propint

BZ

d2k

(2π)2



int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)+

int

BZ

d2k

(2π)2

m2

(P 2(k) +m2)(P 2(k minus q) +m2)


=

int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)

=

int

BZ

d2k

(2π)2


(k2 +m2)(P (k minus q)2 +m2)

minus 2Λsum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2

P micro(k minus q)



sum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2


le 2Cπ2

int minusΛ+q1

minusΛ

dk1

2π

int Λ

minusΛ

dk2

2π

1


=C

2

∣

∣

∣

∣

int Λ

minusΛ


)

ω(k2)minus1

∣

∣

∣

∣


le C

2

int Λ

minusΛ

dk2

∣

∣

∣

q1m2 + k2

2 minus Λq1 + Λ2

∣

∣







52

Introduction


Lattice models












Ward identities









Algorithmic aspects

Quantum Mechanics




Summary and outlook






(3)bos =

a2

2

sum

xy


ϕ =

(

ϕ1

ϕ2

) and ξ =

(

ξ1

ξ2

)


2mφ2 + 1


W prime(φ) = mϕ1 + u+ i(mϕ2 + v) (80)32



(n)0 +m)ϕ +

(

u

v


S(n)bos =

a2

2

sum

xy

(ϕx K(n)xy ϕy) +

a2

2

sum

x


sum

x

ux

(

(div ϕ)x minusar

2(∆ϕ1)x

)

minus a2sum

x

vx

(

(13url ϕ)x +ar

2(∆ϕ2)x

)

∆(2) = a2sum

x

ux

(

(div ϕ)x minusar

2(∆ϕ2)x

)

+ a2sum

x

vx

(


2(∆ϕ1)x

)

(84)∆(3) = a2

sum

x


x



1

2mφ2 +

1


m1minusloop = m(

1 minus g2

4πm2






m


002 004 006 008 01 012 01466577588599510105


35


partH


partφx (87)where

H =1

2

sum

x




prod

x

(

1 +m+ 3gφ2x

)


partM [φ]xypartφz




37


g=2g=1

g=05g=02g=01

R

p(R

)

50-5-10-15-20-25

25

2

15

1

05


quencheddynamical

t

Gbos(t

)

35302520151050

0045

004

0035

003

0025

002

0015

001

0005


39




(

G(x)

G(x+ a)

)

G = G(n)bos or G


G(x) =sum



Gferm(x) =sum

λ


λ (94)40




Gferm(x) =1

N

sum

k

e2πiNkx

2πiNk +m


latti13e point x

〈ψxψ

0〉


0 10 20 30 40 50 60-010010203040506070809


1e-09

1e-08

1e-07

1e-06

1e-05

1e-04

0001

001

01

1

0 20 40 60 80 100 120


latti13e point x

|〈ψxψ

0〉|



1

NtN2x

sum

tprime

sum

xxprime


42

andGferm(t) =

1

NtN2x

2sum

α=1

sum

tprime

sum

xxprime



1









1


1



Iε + Iπ =

int

BZ

d2k

(2π)2

1


Iε =

int

D

d2k

(2π)2

1

(k2 +m2)((k + q1)2 +m2) ((k + qnminus1)2 +m2)

Iπ =

int

BZD

d2k

(2π)2

1


le (Ca2)nminus1

int

|k|leradic

2πa

d2k

(2π)2

1


(

1 +2π2

a2m2

)



int

BZ

d2k

(2π)2



I primeε =

int

D

d2k

(2π)2


I primeπ =

int

BZD

d2k

(2π)2

Pmicro(k) P(k + ql)


leint

BZD

d2k

(2π)2


le(π

a

)l+1

(Ca2)nminus1

int

|k|leradic

2πa

d2k

(2π)2

1


(

1 +2π2

a2m2

)



iar

2γ3∆ with r2 =

4



47

internal linesq

k

propint

BZ

d2k

(2π)2



int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)+

int

BZ

d2k

(2π)2

m2

(P 2(k) +m2)(P 2(k minus q) +m2)


=

int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)

=

int

BZ

d2k

(2π)2


(k2 +m2)(P (k minus q)2 +m2)

minus 2Λsum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2

P micro(k minus q)



sum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2


le 2Cπ2

int minusΛ+q1

minusΛ

dk1

2π

int Λ

minusΛ

dk2

2π

1


=C

2

∣

∣

∣

∣

int Λ

minusΛ


)

ω(k2)minus1

∣

∣

∣

∣


le C

2

int Λ

minusΛ

dk2

∣

∣

∣

q1m2 + k2

2 minus Λq1 + Λ2

∣

∣







52

Introduction


Lattice models












Ward identities









Algorithmic aspects

Quantum Mechanics




Summary and outlook




(n)0 +m)ϕ +

(

u

v


S(n)bos =

a2

2

sum

xy

(ϕx K(n)xy ϕy) +

a2

2

sum

x


sum

x

ux

(

(div ϕ)x minusar

2(∆ϕ1)x

)

minus a2sum

x

vx

(

(13url ϕ)x +ar

2(∆ϕ2)x

)

∆(2) = a2sum

x

ux

(

(div ϕ)x minusar

2(∆ϕ2)x

)

+ a2sum

x

vx

(


2(∆ϕ1)x

)

(84)∆(3) = a2

sum

x


x



1

2mφ2 +

1


m1minusloop = m(

1 minus g2

4πm2






m


002 004 006 008 01 012 01466577588599510105


35


partH


partφx (87)where

H =1

2

sum

x




prod

x

(

1 +m+ 3gφ2x

)


partM [φ]xypartφz




37


g=2g=1

g=05g=02g=01

R

p(R

)

50-5-10-15-20-25

25

2

15

1

05


quencheddynamical

t

Gbos(t

)

35302520151050

0045

004

0035

003

0025

002

0015

001

0005


39




(

G(x)

G(x+ a)

)

G = G(n)bos or G


G(x) =sum



Gferm(x) =sum

λ


λ (94)40




Gferm(x) =1

N

sum

k

e2πiNkx

2πiNk +m


latti13e point x

〈ψxψ

0〉


0 10 20 30 40 50 60-010010203040506070809


1e-09

1e-08

1e-07

1e-06

1e-05

1e-04

0001

001

01

1

0 20 40 60 80 100 120


latti13e point x

|〈ψxψ

0〉|



1

NtN2x

sum

tprime

sum

xxprime


42

andGferm(t) =

1

NtN2x

2sum

α=1

sum

tprime

sum

xxprime



1









1


1



Iε + Iπ =

int

BZ

d2k

(2π)2

1


Iε =

int

D

d2k

(2π)2

1

(k2 +m2)((k + q1)2 +m2) ((k + qnminus1)2 +m2)

Iπ =

int

BZD

d2k

(2π)2

1


le (Ca2)nminus1

int

|k|leradic

2πa

d2k

(2π)2

1


(

1 +2π2

a2m2

)



int

BZ

d2k

(2π)2



I primeε =

int

D

d2k

(2π)2


I primeπ =

int

BZD

d2k

(2π)2

Pmicro(k) P(k + ql)


leint

BZD

d2k

(2π)2


le(π

a

)l+1

(Ca2)nminus1

int

|k|leradic

2πa

d2k

(2π)2

1


(

1 +2π2

a2m2

)



iar

2γ3∆ with r2 =

4



47

internal linesq

k

propint

BZ

d2k

(2π)2



int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)+

int

BZ

d2k

(2π)2

m2

(P 2(k) +m2)(P 2(k minus q) +m2)


=

int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)

=

int

BZ

d2k

(2π)2


(k2 +m2)(P (k minus q)2 +m2)

minus 2Λsum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2

P micro(k minus q)



sum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2


le 2Cπ2

int minusΛ+q1

minusΛ

dk1

2π

int Λ

minusΛ

dk2

2π

1


=C

2

∣

∣

∣

∣

int Λ

minusΛ


)

ω(k2)minus1

∣

∣

∣

∣


le C

2

int Λ

minusΛ

dk2

∣

∣

∣

q1m2 + k2

2 minus Λq1 + Λ2

∣

∣







52

Introduction


Lattice models












Ward identities









Algorithmic aspects

Quantum Mechanics




Summary and outlook



1

2mφ2 +

1


m1minusloop = m(

1 minus g2

4πm2






m


002 004 006 008 01 012 01466577588599510105


35


partH


partφx (87)where

H =1

2

sum

x




prod

x

(

1 +m+ 3gφ2x

)


partM [φ]xypartφz




37


g=2g=1

g=05g=02g=01

R

p(R

)

50-5-10-15-20-25

25

2

15

1

05


quencheddynamical

t

Gbos(t

)

35302520151050

0045

004

0035

003

0025

002

0015

001

0005


39




(

G(x)

G(x+ a)

)

G = G(n)bos or G


G(x) =sum



Gferm(x) =sum

λ


λ (94)40




Gferm(x) =1

N

sum

k

e2πiNkx

2πiNk +m


latti13e point x

〈ψxψ

0〉


0 10 20 30 40 50 60-010010203040506070809


1e-09

1e-08

1e-07

1e-06

1e-05

1e-04

0001

001

01

1

0 20 40 60 80 100 120


latti13e point x

|〈ψxψ

0〉|



1

NtN2x

sum

tprime

sum

xxprime


42

andGferm(t) =

1

NtN2x

2sum

α=1

sum

tprime

sum

xxprime



1









1


1



Iε + Iπ =

int

BZ

d2k

(2π)2

1


Iε =

int

D

d2k

(2π)2

1

(k2 +m2)((k + q1)2 +m2) ((k + qnminus1)2 +m2)

Iπ =

int

BZD

d2k

(2π)2

1


le (Ca2)nminus1

int

|k|leradic

2πa

d2k

(2π)2

1


(

1 +2π2

a2m2

)



int

BZ

d2k

(2π)2



I primeε =

int

D

d2k

(2π)2


I primeπ =

int

BZD

d2k

(2π)2

Pmicro(k) P(k + ql)


leint

BZD

d2k

(2π)2


le(π

a

)l+1

(Ca2)nminus1

int

|k|leradic

2πa

d2k

(2π)2

1


(

1 +2π2

a2m2

)



iar

2γ3∆ with r2 =

4



47

internal linesq

k

propint

BZ

d2k

(2π)2



int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)+

int

BZ

d2k

(2π)2

m2

(P 2(k) +m2)(P 2(k minus q) +m2)


=

int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)

=

int

BZ

d2k

(2π)2


(k2 +m2)(P (k minus q)2 +m2)

minus 2Λsum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2

P micro(k minus q)



sum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2


le 2Cπ2

int minusΛ+q1

minusΛ

dk1

2π

int Λ

minusΛ

dk2

2π

1


=C

2

∣

∣

∣

∣

int Λ

minusΛ


)

ω(k2)minus1

∣

∣

∣

∣


le C

2

int Λ

minusΛ

dk2

∣

∣

∣

q1m2 + k2

2 minus Λq1 + Λ2

∣

∣







52

Introduction


Lattice models












Ward identities









Algorithmic aspects

Quantum Mechanics




Summary and outlook






m


002 004 006 008 01 012 01466577588599510105


35


partH


partφx (87)where

H =1

2

sum

x




prod

x

(

1 +m+ 3gφ2x

)


partM [φ]xypartφz




37


g=2g=1

g=05g=02g=01

R

p(R

)

50-5-10-15-20-25

25

2

15

1

05


quencheddynamical

t

Gbos(t

)

35302520151050

0045

004

0035

003

0025

002

0015

001

0005


39




(

G(x)

G(x+ a)

)

G = G(n)bos or G


G(x) =sum



Gferm(x) =sum

λ


λ (94)40




Gferm(x) =1

N

sum

k

e2πiNkx

2πiNk +m


latti13e point x

〈ψxψ

0〉


0 10 20 30 40 50 60-010010203040506070809


1e-09

1e-08

1e-07

1e-06

1e-05

1e-04

0001

001

01

1

0 20 40 60 80 100 120


latti13e point x

|〈ψxψ

0〉|



1

NtN2x

sum

tprime

sum

xxprime


42

andGferm(t) =

1

NtN2x

2sum

α=1

sum

tprime

sum

xxprime



1









1


1



Iε + Iπ =

int

BZ

d2k

(2π)2

1


Iε =

int

D

d2k

(2π)2

1

(k2 +m2)((k + q1)2 +m2) ((k + qnminus1)2 +m2)

Iπ =

int

BZD

d2k

(2π)2

1


le (Ca2)nminus1

int

|k|leradic

2πa

d2k

(2π)2

1


(

1 +2π2

a2m2

)



int

BZ

d2k

(2π)2



I primeε =

int

D

d2k

(2π)2


I primeπ =

int

BZD

d2k

(2π)2

Pmicro(k) P(k + ql)


leint

BZD

d2k

(2π)2


le(π

a

)l+1

(Ca2)nminus1

int

|k|leradic

2πa

d2k

(2π)2

1


(

1 +2π2

a2m2

)



iar

2γ3∆ with r2 =

4



47

internal linesq

k

propint

BZ

d2k

(2π)2



int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)+

int

BZ

d2k

(2π)2

m2

(P 2(k) +m2)(P 2(k minus q) +m2)


=

int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)

=

int

BZ

d2k

(2π)2


(k2 +m2)(P (k minus q)2 +m2)

minus 2Λsum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2

P micro(k minus q)



sum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2


le 2Cπ2

int minusΛ+q1

minusΛ

dk1

2π

int Λ

minusΛ

dk2

2π

1


=C

2

∣

∣

∣

∣

int Λ

minusΛ


)

ω(k2)minus1

∣

∣

∣

∣


le C

2

int Λ

minusΛ

dk2

∣

∣

∣

q1m2 + k2

2 minus Λq1 + Λ2

∣

∣







52

Introduction


Lattice models












Ward identities









Algorithmic aspects

Quantum Mechanics




Summary and outlook



partH


partφx (87)where

H =1

2

sum

x




prod

x

(

1 +m+ 3gφ2x

)


partM [φ]xypartφz




37


g=2g=1

g=05g=02g=01

R

p(R

)

50-5-10-15-20-25

25

2

15

1

05


quencheddynamical

t

Gbos(t

)

35302520151050

0045

004

0035

003

0025

002

0015

001

0005


39




(

G(x)

G(x+ a)

)

G = G(n)bos or G


G(x) =sum



Gferm(x) =sum

λ


λ (94)40




Gferm(x) =1

N

sum

k

e2πiNkx

2πiNk +m


latti13e point x

〈ψxψ

0〉


0 10 20 30 40 50 60-010010203040506070809


1e-09

1e-08

1e-07

1e-06

1e-05

1e-04

0001

001

01

1

0 20 40 60 80 100 120


latti13e point x

|〈ψxψ

0〉|



1

NtN2x

sum

tprime

sum

xxprime


42

andGferm(t) =

1

NtN2x

2sum

α=1

sum

tprime

sum

xxprime



1









1


1



Iε + Iπ =

int

BZ

d2k

(2π)2

1


Iε =

int

D

d2k

(2π)2

1

(k2 +m2)((k + q1)2 +m2) ((k + qnminus1)2 +m2)

Iπ =

int

BZD

d2k

(2π)2

1


le (Ca2)nminus1

int

|k|leradic

2πa

d2k

(2π)2

1


(

1 +2π2

a2m2

)



int

BZ

d2k

(2π)2



I primeε =

int

D

d2k

(2π)2


I primeπ =

int

BZD

d2k

(2π)2

Pmicro(k) P(k + ql)


leint

BZD

d2k

(2π)2


le(π

a

)l+1

(Ca2)nminus1

int

|k|leradic

2πa

d2k

(2π)2

1


(

1 +2π2

a2m2

)



iar

2γ3∆ with r2 =

4



47

internal linesq

k

propint

BZ

d2k

(2π)2



int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)+

int

BZ

d2k

(2π)2

m2

(P 2(k) +m2)(P 2(k minus q) +m2)


=

int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)

=

int

BZ

d2k

(2π)2


(k2 +m2)(P (k minus q)2 +m2)

minus 2Λsum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2

P micro(k minus q)



sum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2


le 2Cπ2

int minusΛ+q1

minusΛ

dk1

2π

int Λ

minusΛ

dk2

2π

1


=C

2

∣

∣

∣

∣

int Λ

minusΛ


)

ω(k2)minus1

∣

∣

∣

∣


le C

2

int Λ

minusΛ

dk2

∣

∣

∣

q1m2 + k2

2 minus Λq1 + Λ2

∣

∣







52

Introduction


Lattice models












Ward identities









Algorithmic aspects

Quantum Mechanics




Summary and outlook



prod

x

(

1 +m+ 3gφ2x

)


partM [φ]xypartφz




37


g=2g=1

g=05g=02g=01

R

p(R

)

50-5-10-15-20-25

25

2

15

1

05


quencheddynamical

t

Gbos(t

)

35302520151050

0045

004

0035

003

0025

002

0015

001

0005


39




(

G(x)

G(x+ a)

)

G = G(n)bos or G


G(x) =sum



Gferm(x) =sum

λ


λ (94)40




Gferm(x) =1

N

sum

k

e2πiNkx

2πiNk +m


latti13e point x

〈ψxψ

0〉


0 10 20 30 40 50 60-010010203040506070809


1e-09

1e-08

1e-07

1e-06

1e-05

1e-04

0001

001

01

1

0 20 40 60 80 100 120


latti13e point x

|〈ψxψ

0〉|



1

NtN2x

sum

tprime

sum

xxprime


42

andGferm(t) =

1

NtN2x

2sum

α=1

sum

tprime

sum

xxprime



1









1


1



Iε + Iπ =

int

BZ

d2k

(2π)2

1


Iε =

int

D

d2k

(2π)2

1

(k2 +m2)((k + q1)2 +m2) ((k + qnminus1)2 +m2)

Iπ =

int

BZD

d2k

(2π)2

1


le (Ca2)nminus1

int

|k|leradic

2πa

d2k

(2π)2

1


(

1 +2π2

a2m2

)



int

BZ

d2k

(2π)2



I primeε =

int

D

d2k

(2π)2


I primeπ =

int

BZD

d2k

(2π)2

Pmicro(k) P(k + ql)


leint

BZD

d2k

(2π)2


le(π

a

)l+1

(Ca2)nminus1

int

|k|leradic

2πa

d2k

(2π)2

1


(

1 +2π2

a2m2

)



iar

2γ3∆ with r2 =

4



47

internal linesq

k

propint

BZ

d2k

(2π)2



int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)+

int

BZ

d2k

(2π)2

m2

(P 2(k) +m2)(P 2(k minus q) +m2)


=

int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)

=

int

BZ

d2k

(2π)2


(k2 +m2)(P (k minus q)2 +m2)

minus 2Λsum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2

P micro(k minus q)



sum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2


le 2Cπ2

int minusΛ+q1

minusΛ

dk1

2π

int Λ

minusΛ

dk2

2π

1


=C

2

∣

∣

∣

∣

int Λ

minusΛ


)

ω(k2)minus1

∣

∣

∣

∣


le C

2

int Λ

minusΛ

dk2

∣

∣

∣

q1m2 + k2

2 minus Λq1 + Λ2

∣

∣







52

Introduction


Lattice models












Ward identities









Algorithmic aspects

Quantum Mechanics




Summary and outlook



g=2g=1

g=05g=02g=01

R

p(R

)

50-5-10-15-20-25

25

2

15

1

05


quencheddynamical

t

Gbos(t

)

35302520151050

0045

004

0035

003

0025

002

0015

001

0005


39




(

G(x)

G(x+ a)

)

G = G(n)bos or G


G(x) =sum



Gferm(x) =sum

λ


λ (94)40




Gferm(x) =1

N

sum

k

e2πiNkx

2πiNk +m


latti13e point x

〈ψxψ

0〉


0 10 20 30 40 50 60-010010203040506070809


1e-09

1e-08

1e-07

1e-06

1e-05

1e-04

0001

001

01

1

0 20 40 60 80 100 120


latti13e point x

|〈ψxψ

0〉|



1

NtN2x

sum

tprime

sum

xxprime


42

andGferm(t) =

1

NtN2x

2sum

α=1

sum

tprime

sum

xxprime



1









1


1



Iε + Iπ =

int

BZ

d2k

(2π)2

1


Iε =

int

D

d2k

(2π)2

1

(k2 +m2)((k + q1)2 +m2) ((k + qnminus1)2 +m2)

Iπ =

int

BZD

d2k

(2π)2

1


le (Ca2)nminus1

int

|k|leradic

2πa

d2k

(2π)2

1


(

1 +2π2

a2m2

)



int

BZ

d2k

(2π)2



I primeε =

int

D

d2k

(2π)2


I primeπ =

int

BZD

d2k

(2π)2

Pmicro(k) P(k + ql)


leint

BZD

d2k

(2π)2


le(π

a

)l+1

(Ca2)nminus1

int

|k|leradic

2πa

d2k

(2π)2

1


(

1 +2π2

a2m2

)



iar

2γ3∆ with r2 =

4



47

internal linesq

k

propint

BZ

d2k

(2π)2



int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)+

int

BZ

d2k

(2π)2

m2

(P 2(k) +m2)(P 2(k minus q) +m2)


=

int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)

=

int

BZ

d2k

(2π)2


(k2 +m2)(P (k minus q)2 +m2)

minus 2Λsum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2

P micro(k minus q)



sum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2


le 2Cπ2

int minusΛ+q1

minusΛ

dk1

2π

int Λ

minusΛ

dk2

2π

1


=C

2

∣

∣

∣

∣

int Λ

minusΛ


)

ω(k2)minus1

∣

∣

∣

∣


le C

2

int Λ

minusΛ

dk2

∣

∣

∣

q1m2 + k2

2 minus Λq1 + Λ2

∣

∣







52

Introduction


Lattice models












Ward identities









Algorithmic aspects

Quantum Mechanics




Summary and outlook


quencheddynamical

t

Gbos(t

)

35302520151050

0045

004

0035

003

0025

002

0015

001

0005


39




(

G(x)

G(x+ a)

)

G = G(n)bos or G


G(x) =sum



Gferm(x) =sum

λ


λ (94)40




Gferm(x) =1

N

sum

k

e2πiNkx

2πiNk +m


latti13e point x

〈ψxψ

0〉


0 10 20 30 40 50 60-010010203040506070809


1e-09

1e-08

1e-07

1e-06

1e-05

1e-04

0001

001

01

1

0 20 40 60 80 100 120


latti13e point x

|〈ψxψ

0〉|



1

NtN2x

sum

tprime

sum

xxprime


42

andGferm(t) =

1

NtN2x

2sum

α=1

sum

tprime

sum

xxprime



1









1


1



Iε + Iπ =

int

BZ

d2k

(2π)2

1


Iε =

int

D

d2k

(2π)2

1

(k2 +m2)((k + q1)2 +m2) ((k + qnminus1)2 +m2)

Iπ =

int

BZD

d2k

(2π)2

1


le (Ca2)nminus1

int

|k|leradic

2πa

d2k

(2π)2

1


(

1 +2π2

a2m2

)



int

BZ

d2k

(2π)2



I primeε =

int

D

d2k

(2π)2


I primeπ =

int

BZD

d2k

(2π)2

Pmicro(k) P(k + ql)


leint

BZD

d2k

(2π)2


le(π

a

)l+1

(Ca2)nminus1

int

|k|leradic

2πa

d2k

(2π)2

1


(

1 +2π2

a2m2

)



iar

2γ3∆ with r2 =

4



47

internal linesq

k

propint

BZ

d2k

(2π)2



int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)+

int

BZ

d2k

(2π)2

m2

(P 2(k) +m2)(P 2(k minus q) +m2)


=

int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)

=

int

BZ

d2k

(2π)2


(k2 +m2)(P (k minus q)2 +m2)

minus 2Λsum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2

P micro(k minus q)



sum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2


le 2Cπ2

int minusΛ+q1

minusΛ

dk1

2π

int Λ

minusΛ

dk2

2π

1


=C

2

∣

∣

∣

∣

int Λ

minusΛ


)

ω(k2)minus1

∣

∣

∣

∣


le C

2

int Λ

minusΛ

dk2

∣

∣

∣

q1m2 + k2

2 minus Λq1 + Λ2

∣

∣







52

Introduction


Lattice models












Ward identities









Algorithmic aspects

Quantum Mechanics




Summary and outlook





(

G(x)

G(x+ a)

)

G = G(n)bos or G


G(x) =sum



Gferm(x) =sum

λ


λ (94)40




Gferm(x) =1

N

sum

k

e2πiNkx

2πiNk +m


latti13e point x

〈ψxψ

0〉


0 10 20 30 40 50 60-010010203040506070809


1e-09

1e-08

1e-07

1e-06

1e-05

1e-04

0001

001

01

1

0 20 40 60 80 100 120


latti13e point x

|〈ψxψ

0〉|



1

NtN2x

sum

tprime

sum

xxprime


42

andGferm(t) =

1

NtN2x

2sum

α=1

sum

tprime

sum

xxprime



1









1


1



Iε + Iπ =

int

BZ

d2k

(2π)2

1


Iε =

int

D

d2k

(2π)2

1

(k2 +m2)((k + q1)2 +m2) ((k + qnminus1)2 +m2)

Iπ =

int

BZD

d2k

(2π)2

1


le (Ca2)nminus1

int

|k|leradic

2πa

d2k

(2π)2

1


(

1 +2π2

a2m2

)



int

BZ

d2k

(2π)2



I primeε =

int

D

d2k

(2π)2


I primeπ =

int

BZD

d2k

(2π)2

Pmicro(k) P(k + ql)


leint

BZD

d2k

(2π)2


le(π

a

)l+1

(Ca2)nminus1

int

|k|leradic

2πa

d2k

(2π)2

1


(

1 +2π2

a2m2

)



iar

2γ3∆ with r2 =

4



47

internal linesq

k

propint

BZ

d2k

(2π)2



int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)+

int

BZ

d2k

(2π)2

m2

(P 2(k) +m2)(P 2(k minus q) +m2)


=

int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)

=

int

BZ

d2k

(2π)2


(k2 +m2)(P (k minus q)2 +m2)

minus 2Λsum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2

P micro(k minus q)



sum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2


le 2Cπ2

int minusΛ+q1

minusΛ

dk1

2π

int Λ

minusΛ

dk2

2π

1


=C

2

∣

∣

∣

∣

int Λ

minusΛ


)

ω(k2)minus1

∣

∣

∣

∣


le C

2

int Λ

minusΛ

dk2

∣

∣

∣

q1m2 + k2

2 minus Λq1 + Λ2

∣

∣







52

Introduction


Lattice models












Ward identities









Algorithmic aspects

Quantum Mechanics




Summary and outlook





Gferm(x) =1

N

sum

k

e2πiNkx

2πiNk +m


latti13e point x

〈ψxψ

0〉


0 10 20 30 40 50 60-010010203040506070809


1e-09

1e-08

1e-07

1e-06

1e-05

1e-04

0001

001

01

1

0 20 40 60 80 100 120


latti13e point x

|〈ψxψ

0〉|



1

NtN2x

sum

tprime

sum

xxprime


42

andGferm(t) =

1

NtN2x

2sum

α=1

sum

tprime

sum

xxprime



1









1


1



Iε + Iπ =

int

BZ

d2k

(2π)2

1


Iε =

int

D

d2k

(2π)2

1

(k2 +m2)((k + q1)2 +m2) ((k + qnminus1)2 +m2)

Iπ =

int

BZD

d2k

(2π)2

1


le (Ca2)nminus1

int

|k|leradic

2πa

d2k

(2π)2

1


(

1 +2π2

a2m2

)



int

BZ

d2k

(2π)2



I primeε =

int

D

d2k

(2π)2


I primeπ =

int

BZD

d2k

(2π)2

Pmicro(k) P(k + ql)


leint

BZD

d2k

(2π)2


le(π

a

)l+1

(Ca2)nminus1

int

|k|leradic

2πa

d2k

(2π)2

1


(

1 +2π2

a2m2

)



iar

2γ3∆ with r2 =

4



47

internal linesq

k

propint

BZ

d2k

(2π)2



int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)+

int

BZ

d2k

(2π)2

m2

(P 2(k) +m2)(P 2(k minus q) +m2)


=

int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)

=

int

BZ

d2k

(2π)2


(k2 +m2)(P (k minus q)2 +m2)

minus 2Λsum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2

P micro(k minus q)



sum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2


le 2Cπ2

int minusΛ+q1

minusΛ

dk1

2π

int Λ

minusΛ

dk2

2π

1


=C

2

∣

∣

∣

∣

int Λ

minusΛ


)

ω(k2)minus1

∣

∣

∣

∣


le C

2

int Λ

minusΛ

dk2

∣

∣

∣

q1m2 + k2

2 minus Λq1 + Λ2

∣

∣







52

Introduction


Lattice models












Ward identities









Algorithmic aspects

Quantum Mechanics




Summary and outlook


1e-09

1e-08

1e-07

1e-06

1e-05

1e-04

0001

001

01

1

0 20 40 60 80 100 120


latti13e point x

|〈ψxψ

0〉|



1

NtN2x

sum

tprime

sum

xxprime


42

andGferm(t) =

1

NtN2x

2sum

α=1

sum

tprime

sum

xxprime



1









1


1



Iε + Iπ =

int

BZ

d2k

(2π)2

1


Iε =

int

D

d2k

(2π)2

1

(k2 +m2)((k + q1)2 +m2) ((k + qnminus1)2 +m2)

Iπ =

int

BZD

d2k

(2π)2

1


le (Ca2)nminus1

int

|k|leradic

2πa

d2k

(2π)2

1


(

1 +2π2

a2m2

)



int

BZ

d2k

(2π)2



I primeε =

int

D

d2k

(2π)2


I primeπ =

int

BZD

d2k

(2π)2

Pmicro(k) P(k + ql)


leint

BZD

d2k

(2π)2


le(π

a

)l+1

(Ca2)nminus1

int

|k|leradic

2πa

d2k

(2π)2

1


(

1 +2π2

a2m2

)



iar

2γ3∆ with r2 =

4



47

internal linesq

k

propint

BZ

d2k

(2π)2



int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)+

int

BZ

d2k

(2π)2

m2

(P 2(k) +m2)(P 2(k minus q) +m2)


=

int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)

=

int

BZ

d2k

(2π)2


(k2 +m2)(P (k minus q)2 +m2)

minus 2Λsum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2

P micro(k minus q)



sum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2


le 2Cπ2

int minusΛ+q1

minusΛ

dk1

2π

int Λ

minusΛ

dk2

2π

1


=C

2

∣

∣

∣

∣

int Λ

minusΛ


)

ω(k2)minus1

∣

∣

∣

∣


le C

2

int Λ

minusΛ

dk2

∣

∣

∣

q1m2 + k2

2 minus Λq1 + Λ2

∣

∣







52

Introduction


Lattice models












Ward identities









Algorithmic aspects

Quantum Mechanics




Summary and outlook


andGferm(t) =

1

NtN2x

2sum

α=1

sum

tprime

sum

xxprime



1









1


1



Iε + Iπ =

int

BZ

d2k

(2π)2

1


Iε =

int

D

d2k

(2π)2

1

(k2 +m2)((k + q1)2 +m2) ((k + qnminus1)2 +m2)

Iπ =

int

BZD

d2k

(2π)2

1


le (Ca2)nminus1

int

|k|leradic

2πa

d2k

(2π)2

1


(

1 +2π2

a2m2

)



int

BZ

d2k

(2π)2



I primeε =

int

D

d2k

(2π)2


I primeπ =

int

BZD

d2k

(2π)2

Pmicro(k) P(k + ql)


leint

BZD

d2k

(2π)2


le(π

a

)l+1

(Ca2)nminus1

int

|k|leradic

2πa

d2k

(2π)2

1


(

1 +2π2

a2m2

)



iar

2γ3∆ with r2 =

4



47

internal linesq

k

propint

BZ

d2k

(2π)2



int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)+

int

BZ

d2k

(2π)2

m2

(P 2(k) +m2)(P 2(k minus q) +m2)


=

int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)

=

int

BZ

d2k

(2π)2


(k2 +m2)(P (k minus q)2 +m2)

minus 2Λsum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2

P micro(k minus q)



sum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2


le 2Cπ2

int minusΛ+q1

minusΛ

dk1

2π

int Λ

minusΛ

dk2

2π

1


=C

2

∣

∣

∣

∣

int Λ

minusΛ


)

ω(k2)minus1

∣

∣

∣

∣


le C

2

int Λ

minusΛ

dk2

∣

∣

∣

q1m2 + k2

2 minus Λq1 + Λ2

∣

∣







52

Introduction


Lattice models












Ward identities









Algorithmic aspects

Quantum Mechanics




Summary and outlook







1


1



Iε + Iπ =

int

BZ

d2k

(2π)2

1


Iε =

int

D

d2k

(2π)2

1

(k2 +m2)((k + q1)2 +m2) ((k + qnminus1)2 +m2)

Iπ =

int

BZD

d2k

(2π)2

1


le (Ca2)nminus1

int

|k|leradic

2πa

d2k

(2π)2

1


(

1 +2π2

a2m2

)



int

BZ

d2k

(2π)2



I primeε =

int

D

d2k

(2π)2


I primeπ =

int

BZD

d2k

(2π)2

Pmicro(k) P(k + ql)


leint

BZD

d2k

(2π)2


le(π

a

)l+1

(Ca2)nminus1

int

|k|leradic

2πa

d2k

(2π)2

1


(

1 +2π2

a2m2

)



iar

2γ3∆ with r2 =

4



47

internal linesq

k

propint

BZ

d2k

(2π)2



int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)+

int

BZ

d2k

(2π)2

m2

(P 2(k) +m2)(P 2(k minus q) +m2)


=

int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)

=

int

BZ

d2k

(2π)2


(k2 +m2)(P (k minus q)2 +m2)

minus 2Λsum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2

P micro(k minus q)



sum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2


le 2Cπ2

int minusΛ+q1

minusΛ

dk1

2π

int Λ

minusΛ

dk2

2π

1


=C

2

∣

∣

∣

∣

int Λ

minusΛ


)

ω(k2)minus1

∣

∣

∣

∣


le C

2

int Λ

minusΛ

dk2

∣

∣

∣

q1m2 + k2

2 minus Λq1 + Λ2

∣

∣







52

Introduction


Lattice models












Ward identities









Algorithmic aspects

Quantum Mechanics




Summary and outlook



int

BZ

d2k

(2π)2



I primeε =

int

D

d2k

(2π)2


I primeπ =

int

BZD

d2k

(2π)2

Pmicro(k) P(k + ql)


leint

BZD

d2k

(2π)2


le(π

a

)l+1

(Ca2)nminus1

int

|k|leradic

2πa

d2k

(2π)2

1


(

1 +2π2

a2m2

)



iar

2γ3∆ with r2 =

4



47

internal linesq

k

propint

BZ

d2k

(2π)2



int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)+

int

BZ

d2k

(2π)2

m2

(P 2(k) +m2)(P 2(k minus q) +m2)


=

int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)

=

int

BZ

d2k

(2π)2


(k2 +m2)(P (k minus q)2 +m2)

minus 2Λsum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2

P micro(k minus q)



sum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2


le 2Cπ2

int minusΛ+q1

minusΛ

dk1

2π

int Λ

minusΛ

dk2

2π

1


=C

2

∣

∣

∣

∣

int Λ

minusΛ


)

ω(k2)minus1

∣

∣

∣

∣


le C

2

int Λ

minusΛ

dk2

∣

∣

∣

q1m2 + k2

2 minus Λq1 + Λ2

∣

∣







52

Introduction


Lattice models












Ward identities









Algorithmic aspects

Quantum Mechanics




Summary and outlook



iar

2γ3∆ with r2 =

4



47

internal linesq

k

propint

BZ

d2k

(2π)2



int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)+

int

BZ

d2k

(2π)2

m2

(P 2(k) +m2)(P 2(k minus q) +m2)


=

int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)

=

int

BZ

d2k

(2π)2


(k2 +m2)(P (k minus q)2 +m2)

minus 2Λsum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2

P micro(k minus q)



sum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2


le 2Cπ2

int minusΛ+q1

minusΛ

dk1

2π

int Λ

minusΛ

dk2

2π

1


=C

2

∣

∣

∣

∣

int Λ

minusΛ


)

ω(k2)minus1

∣

∣

∣

∣


le C

2

int Λ

minusΛ

dk2

∣

∣

∣

q1m2 + k2

2 minus Λq1 + Λ2

∣

∣







52

Introduction


Lattice models












Ward identities









Algorithmic aspects

Quantum Mechanics




Summary and outlook



47

internal linesq

k

propint

BZ

d2k

(2π)2



int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)+

int

BZ

d2k

(2π)2

m2

(P 2(k) +m2)(P 2(k minus q) +m2)


=

int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)

=

int

BZ

d2k

(2π)2


(k2 +m2)(P (k minus q)2 +m2)

minus 2Λsum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2

P micro(k minus q)



sum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2


le 2Cπ2

int minusΛ+q1

minusΛ

dk1

2π

int Λ

minusΛ

dk2

2π

1


=C

2

∣

∣

∣

∣

int Λ

minusΛ


)

ω(k2)minus1

∣

∣

∣

∣


le C

2

int Λ

minusΛ

dk2

∣

∣

∣

q1m2 + k2

2 minus Λq1 + Λ2

∣

∣







52

Introduction


Lattice models












Ward identities









Algorithmic aspects

Quantum Mechanics




Summary and outlook


internal linesq

k

propint

BZ

d2k

(2π)2



int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)+

int

BZ

d2k

(2π)2

m2

(P 2(k) +m2)(P 2(k minus q) +m2)


=

int

BZ

d2k

(2π)2


(P 2(k) +m2)(P 2(k minus q) +m2)

=

int

BZ

d2k

(2π)2


(k2 +m2)(P (k minus q)2 +m2)

minus 2Λsum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2

P micro(k minus q)



sum

micro

int minusΛ+qmicro

minusΛ

dkmicro

int Λ

minusΛ

dkν 6=micro(2π)2


le 2Cπ2

int minusΛ+q1

minusΛ

dk1

2π

int Λ

minusΛ

dk2

2π

1


=C

2

∣

∣

∣

∣

int Λ

minusΛ


)

ω(k2)minus1

∣

∣

∣

∣


le C

2

int Λ

minusΛ

dk2

∣

∣

∣

q1m2 + k2

2 minus Λq1 + Λ2

∣

∣







52

Introduction


Lattice models












Ward identities









Algorithmic aspects

Quantum Mechanics




Summary and outlook


le C

2

int Λ

minusΛ

dk2

∣

∣

∣

q1m2 + k2

2 minus Λq1 + Λ2

∣

∣







52

Introduction


Lattice models












Ward identities









Algorithmic aspects

Quantum Mechanics




Summary and outlook






52

Introduction


Lattice models












Ward identities









Algorithmic aspects

Quantum Mechanics




Summary and outlook


N = 2 - uni-jena.de · 11.15.Ha, 11.10.Gh Keyw ords: sup ersymmetry, lattice mo dels 1 Supp orted b...

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Transcript of N = 2 - uni-jena.de · 11.15.Ha, 11.10.Gh Keyw ords: sup ersymmetry, lattice mo dels 1 Supp orted b...