Hallberg & Strykowski - On the Universality of Global Modes in Low-Density Jets (2006)
The density matrix renormalization group · 2018-02-08 · The density-matrix renormaliza1on group,...
Transcript of The density matrix renormalization group · 2018-02-08 · The density-matrix renormaliza1on group,...
Thedensitymatrixrenormaliza2ongroup
Adrian Feiguin
Sourcecode(PythonandC++)
h3p://www.github.com/afeiguin/h3p://www.github.com/afeiguin/comp-phys
ALPSlibrariesandcode:h3p://alps.comp-phys.org
Theselectures:
Someliterature-S.R.White:.Densitymatrixformula1onforquantumrenormaliza1ongroups,Phys.Rev.Le3.69,2863(1992)..Density-matrixalgorithmsforquantumrenormaliza1ongroups,Phys.Rev.B48,10345(1993).-U.Schollwöck.Thedensity-matrixrenormaliza1ongroup,Rev.Mod.Phys.77,259(2005).-KarenHallberg.DensityMatrixRenormaliza1on:AReviewoftheMethodanditsApplica1onsinTheoreYcalMethodsforStronglyCorrelatedElectrons,CRMSeriesinMathemaYcalPhysics,DavidSenechal,Andre-MarieTremblayandClaudeBourbonnais(eds.),Springer,NewYork,2003.NewTrendsinDensityMatrixRenormaliza1on,Adv.Phys.55,477(2006).-The“DMRGBOOK”:Density-MatrixRenormaliza1on-ANewNumericalMethodinPhysics:LecturesofaSeminarandWorkshopheldattheMax-Planck-Ins1tutfürPhysik...18th,1998(LectureNotesinPhysics)byIngoPeschel,XiaoqunWang,Ma3hiasKaulkeandKarenHallberg-R.NoackandS.Manmana:Diagonaliza1on-andNumericalRenormaliza1on-Group-BasedMethodsforInterac1ngQuantumSystemsProceedingsofthe"IX.TrainingCourseinthePhysicsofCorrelatedElectronSystemsandHigh-TcSuperconductors",VietrisulMare(Salerno,Italy,October2004)AIPConf.Proc.789,93-163(2005)-A.E.Feiguin:TheDensityMatrixRenormaliza1onGroupandits1me-dependentvariantsVietriLectureNotes,AIPConferenceProceedings1419,5(2011);h3ps://doi.org/10.1063/1.3667323-F.Verstraete,V.Murg&J.I.Cirac:Matrixproductstates,projectedentangledpairstates,andvariaYonalrenormalizaYongroupmethodsforquantumspinsystems,AdvancesinPhysics,57:2,143-224(2008)-U.Schollwock,Thedensity-matricrenormalizaYongroupintheageofmatricproductstates,AnnalsofPhysics326,96(2011)-FrankPollman’slecturenotes:EfficientNumericalSimulaYonsUsingMatrix-ProductStates
-RomanOrus:ApracYcalintroducYontotensornetworks:Matrixproductstatesandprojectedentangledpairstates,AnnalsofPhysics349(2014)117-158(2014)
Briefhistoryandmilestones• (1992)SteveWhiteintroducestheDMRG.• (1995-…)DynamicalDMRG.(Hallberg,Ramaseshaetal,KuhnerandWhite,
Jeckelmann)• (1995)Nishinointroducesthetransfer-matrixDMRG(TMRG)forclassicalsystems.• (1996-97)Bursill,WangandXiang,Shibata,generalizetheTMRGtoquantum
problems.• (1996)XiangadaptsDMRGtomomentumspace.• (2001)ShibataandYoshiokastudyFQHsystems.• (2004)VidalintroducestheTEBD.(Yme-evolvingblockdecimaYon)• (2005)VerstraeteandCiracintroduceanalternaYvealgorithmforMPS’sand
explainproblemwithDMRGandPBC.• (2006)WhiteandAEF,andDaley,Kollath,Schollwoeck,Vidalgeneralizetheideas
withinaDMRGframework:adapYvetDMRG.
…theDMRGhasbeenusedinavarietyoffieldsandcontexts,fromclassicalsystemstoquantumchemistry,tonuclearphysics…
Singlepar2clevs.manybodypicture
4 configurations 4x4 matrix
EF
The state of the system is a “product state” of single particle states. We only need to solve the one-particle
problem.Think “Hydrogen atom”.
And similarly for the “down” electrons
Singlepar2clevs.manybodypicture
16 configurations 16x16 matrix
The state of the system is “highly entangled”. It cannot be written as a “product state”, and the behavior of each electron is dictated by the behavior of the rest.(Notice that in some case Bethe Ansatz tells us that some many-body states can still be reduced to product states)
Exact diagonalization “brute force” diagonalization of the Hamiltonian matrix.
…anythingyouwanttoknow…but…onlysmallsystems
H |x〉 = E |x〉 H : Hamiltonian operator |x〉 : eigenstate E : eigenvalue (ENERGY)
Schrödinger's Equation:
AllweneedtodoistopickabasisandwritetheHamiltonianmatrixinthatbasis
Symmetries SH=HS
Reflections
Translations
D' = D / N
D' = D / 2
Particle number conservation => Ntotal Spin conservations => Sz
total
Spin reversal => |↑↓〉 ± |↓↑〉
|ψk〉 = (1/M) ∑iaki Ti |φ〉; aki =exp(ikxi)
Block diagonalization
0
0 0
0
EDExample:Heisenbergchain
Geometry:1D chain
Basis:
HHeis = J ∑<i,j> SizSj
z+ 1/2 (Si-Sj
++Si+Sj
-)
ModelHamiltonian:
|↑↓↑↓〉; |↓↑↓↑〉; |↑↑↓↓〉; |↓↑↑↓〉; |↓↓↑↑〉; |↑↓↓↑〉
ApplyingtranslaYons:
|1〉=1/(2√2){(1+ ei2k )|↑↓↑↓〉+ eik(1+ei2k)|↓↑↓↑〉} |2〉=1/2{|↑↑↓↓〉+eik|↓↑↑↓〉+ei2k|↓↓↑↑〉+ei3k|↑↓↓↑〉}
TranslaYons
Withk=0,-π/2,π/2,π
k=0) |1〉=1/√2{|↑↓↑↓〉+|↓↑↓↑〉} |2〉=1/2{|↑↑↓↓〉+|↓↑↑↓〉+|↓↓↑↑〉+|↑↓↓↑〉}
k=-π/2) |2〉=1/2{|↑↑↓↓〉+e-iπ/2|↓↑↑↓〉-|↓↓↑↑〉+eiπ/2 |↑↓↓↑〉}
k=π/2) |2〉=1/2{|↑↑↓↓〉+eiπ/2|↓↑↑↓〉-|↓↓↑↑〉+e-iπ/2 |↑↓↓↑〉}
k=π) |1〉=1/√2{|↑↓↑↓〉-|↓↑↓↑〉} |2〉=1/2{|↑↑↓↓〉-|↓↑↑↓〉+|↓↓↑↑〉-|↑↓↓↑〉}
Limita2ons:smalllaAces• Hubbardmodel:20sitesathalffilling,10↑and10↓,D=20!(10!
10!)x20!(10!10!)=2.4e+10.AuersymmetriesD'=1.1e+8• t-Jmodel(only|o〉, |↑〉 and|↓〉states):32siteswith4holes,
14↑and14↓,D=32!/(14!18!)x18!/(14!4!)=1.4e+12;D'=5.6e9• Heisenbergmodel(only|↑〉 and|↓〉states):36sites,18↑and
18↓,D=36!/(18!18!)=9075135300;D'=D/(36x2x2x2x2)=1.5e6states
Exact diag. is limited by system size… How can we overcome
this problem?
Po’ man’s solution: What about truncating the basis?
“Classical”analogyImagecompressionalgorithms(e.g.Jpeg)
Wewanttoachieve“losslesscompression”…oratleastminimizethelossofinforma1on
Idea 1: Truncated diagonalization
|gs〉 =∑ ai|xi〉
Usually, only a few important states possess most of the weight
, ∑ |ai|2 = 1
Error = 1-∑' |ai|2
Cuthere
Idea 2: Change of basis
Can we rotate our basis to one where the weights are more concentrated, to minimize the error?
|gs〉 =∑ ai|xi〉 , ∑ |ai|2 = 1 Error = 1-∑' |ai|
2
Cuthere Cuthere
Whatdoesitmean“totruncatethebasis”
Ifwetruncate
ThistransformaYonisnolongerunitary,doesnotpreservenorm->lossofinformaYon
Thecaseofspins The two-site basis is given by the states
|σσ’〉 ={|↑↑〉;|↑↓〉;|↓↑〉; |↓↓〉}
We can easily diagonalize the Hamiltonian by rotating with the matrix:
That yields the eigenstates:
Thecaseofspins…
NumericalRenormaliza2onGroup
Let’sconsiderthe1dHeisenbergmodel
( )∑∑ ++
−−+
+++ ++=⋅=
iiiii
zi
zi
iii SSSSSSSSH 1111 2
1!!
⎟⎟⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛
−= −+
1000
;0100
;2/10
02/1000 SSS z
Forasinglesite,theoperatormatricesare:
WealsoneedtodefinetheidenYtyonablockoflsites
22 dimensions with ;10
01ll
lI ×⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
= !
BuildingtheHamiltonianalaNRG
+−−+ ⊗+⊗+⊗= 0000002 21
21 SSSSSSH zz
1 2
ll-1
32 ⎥⎦
⎤⎢⎣
⎡ ⊗+⊗+⊗⊗+⊗= +−−+0000001123 2
121 SSSSSSIIHH zz
221 HIIHH llll ⊗+⊗=→ −−
Thisrecursionwillgeneratea2lx2lHamiltonianmatrixthatwecaneasilydiagonalize
Anotherwaytoputit…
ll-1
( ) ( ) ( )+−
−−+
−−−
+−−
−+−−−
+−−+−−
⊗+⊗+⊗+⊗=
⊗⊗+⊗⊗+⊗⊗+⊗=
⎥⎦
⎤⎢⎣
⎡ ⊗+⊗+⊗⊗+⊗=
01010111
00210200211
000000211
21
21
21
21
21
21
SSSSSSIH
SSISSISSIIH
SSSSSSIIHH
llzz
ll
llzz
ll
zzlll
01 OIO ll ⊗= −with
Addingasinglesitetotheblock
1+lαlα
1+ls
BeforetruncaYngwebuildthenewbasisas:
11 ++ ⊗= lll sαα
AndtheHamiltonianforthenewblockas
...'0,01,1, +⊗+⊗+⊗=+ OOHIIHH lLllLlL
01, OIO llL ⊗= −with
|ψ〉 = ∑ijψij|i〉| j〉
Dim=2L
Dimensionoftheblockgrowsexponen2ally
Idea 3: Density Matrix Renormalization Group S.R. White, Phys. Rev. Lett. 69, 2863(1992), Phys. Rev. B 48, 10345 (1993)
BlockdecimaYon
Dim=2NDim=m
constant
|ψ〉 = ∑ijψij|i〉| j〉
Thedensitymatrixprojec2on
Universe
system
|i〉environment
| j〉
We need to find the transformation
that minimizes the distance
S=||ψ'〉 -|ψ〉|2
|ψ〉 = ∑ijψij|i〉| j〉 |ψ'〉 = ∑mαjaαj|α〉| j〉
Solution: The optimal states are the eigenvectors of the reduced density matrix
ρii' = ∑jψ*ijψi'j Tr ρ = 1
with the m largest eigenvalues ωα
Understandingthedensity-matrixprojecYon
∑=ij
BAijABjiψψ
∑==→
=
jjiijAAAiiA
ABABBA
ii *'' ')(
tr
ψψρρ
ψψρThereduceddensitymatrixisdefinedas:
Universe
system
|i〉environment
| j〉RegionA RegionB
ProperYesofthedensitymatrixψψρ
ABABBA tr=
• HermiYan->eigenvaluesarereal• Eigenvaluesarenon-negaYve• Thetraceequalstounity->TrρA=1• Eigenvectorsformanorthonormalbasis.
1 and 0 with ; =≥= ∑∑α
ααα
α ωωααωρAAA
ThesingularvaluedecomposiYon(SVD)Consideramatrix
ψij= dimA
dimB
WecandecomposeitintotheproductofthreematricesU,D,V:ψ =UDV†
• Uisa(dimAxdimB)matrixwithorthonormalcolumns->UU†=1;U=U†• Disa(dimBxdimB)diagonalmatrixwithnon-negaYveelementsλα• Visa(dimBxdimB)unitarymatrix->VV†=1
= xxψ U D V
(wearechoosingdimB<dimAforconvenience)
( )
( ) ( )
∑
∑∑ ∑∑
∑∑
∑∑
=→
=⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠
⎞⎜⎝
⎛=
=
==
B
BAAB
B
BA
B
jBj
iAi
ij
B
BAjiAB
B
ji
B
jt
iij
jViU
jiVU
VUVU
dim
dimdim*
dim*
dim*
dim
αα
αα
αααα
αααα
αααα
αααα
ααλψ
αλαλ
λψ
λλψ
{ }{ } l!orthonorma are , bases theHereBA
αα
Thisisalsocalledthe“SchmidtdecomposiYon”ofthestate
TheSVDandthedensitymatrix
)dim,min(dim with BA
r
BAABr ==∑
αα ααλψ
Ingeneral:
IntheSchmidtbasis,thereduceddensitymatrixis
∑
∑
=
==
r
BBB
r
AAABABBA
αα
αα
ααλρ
ααλψψρ
2
2
and
tr
• Thesingularvaluesaretheeigenvaluesofthereducedd.m.squaredωi=λi 2
• Thetworeduceddensitymatricessharethespectrum• thesingularvectorsaretheeigenvectorsofthedensitymatrix.
OpYmizingthewave-funcYon
S=||ψ'〉 -|ψ〉|2
Wewanttominimizethedistancebetweenthetwostates
where|ψ〉istheactualgroundstate,and|ψ’〉isthevariaYonalapproximaYonauerrotaYngtoanewbasisandtruncaYng
|ψ'〉 = ∑mαjaαj|α〉| j〉
WereformulatethequesYonas:Givenamatrixψ ,whatistheopYmalmatrixψ’withfixedrankrthatminimizestheFrobeniusdistancebetweenthetwomatrices.Itturnout,thisisawellknownproblem,calledthe“lowrankmatrixapproximaYon”or“pricipalcomponentapproximaYon”(PCA)inmachinelearning.Ifweordertheeigenvaluesofthedensitymatrixindescendingorderω1, ω2,…,ωm,…, ωrweobtain
S=||ψ'〉 -|ψ〉|2 = ∑+
r
mi
1ω TruncaYonerror!
DMRG:TheAlgorithmHowdowebuildthereducedbasisofstates?
Wegrowourbasissystema2cally,addingsitestooursystemateachstep,andusingthedensitymatrixprojec2ontotruncate
2) We diagonalize the system and obtain the ground state |gs〉=∑ψ1234|α1〉|s2〉|s3〉|β4〉
3) We calculate the reduced density matrix ρ for blocks 1-2 and 3-4.
4) We diagonalize ρ obtaining the eigenvectors and eigenvalues ωi
1) We start from a small superblock with 4 sites/blocks, each with a dimension mi , small enough to be easily diagonalized
1 2 3 4 m1
H1
The Algorithm
∑=43
*34'2'1123421121 ''
β
ψψαραs
ss
5) We add a new site to blocks 1 and 4, expanding the basis for each block to m'1 = m m2 and m'4 = m3 m
m2 m m'1=m m2
7) We repeat starting from 2) replacing H1 by H'1 and H4 by H'4
1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4
6) We rotate the Hamiltonian and operators to the new basis keeping the m states with larger eigenvalues (notice that we no longer are in the occupation number representation)
We add one site at a time, until we reach the desired system size
1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4
Thefinitesizealgorithm
1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4
We sweep from left to right We sweep from right to left
Thefinitesizealgorithm
…Until we converge
Finite-sizeDMRGFlowchart
The discarded weight 1- ∑mα=1ωα measures
the accuracy of the truncation to m states
Observa2ons• SweepingisessenYaltoachieveconvergence• Runthefinite-sizeDMRGandextrapolatetothethermodynamiclimit.
• Foreachsystemsize,extrapolatetheresultswiththenumberofstatesm,orfixthetruncaYonerrorbelowcertaintolerance.
DensityMatrixRenormaliza2onGroup
Avaria2onalmethodwithouta-prioriassump2onsaboutthephysics.
• Similarcapabili1esasexactdiagonaliza1on.
• Cancalculateproper1esofverylargesystems(1Dandquasi-2D)withunprecedentedaccuracy.
• Resultsarevaria1onal,but“quasi-exact”:Accuracyisfinite,butundercontrol.
AdvantagesoftheDMRG
• DMRGisveryversaYle,andeasytoadapttocomplexgeometriesandHamiltonians.
• Canbeusedtostudymodelsofspins,bosons,orfermions.
• Generalandreusablecode:Asingleprogramcanbeusedtorunarbitrarymodelswithoutchangingasingleline(e.g.ALPSDMRG)
• Symmetriesareeasytoimplement.
Limita2onsoftheDMRG
• DMRGisthemethodofchoicein1dandquasi-1dsystems,butitislessefficientinhigherdimensions.
• Problemswith(i)criYcalsystems,(ii)longrangeinteracYons,and(iii)periodicboundarycondiYons.
• TheselimitaYonsaredueto:– ThestructureofthevariaYonalwavefuncYonusedbytheDMRG(theMPSansatz).
– Entanglemententropyfollowsarealaw.
TechnicaliYes…Addingasinglesitetotheblock
1+lαlα
1+ls
BeforetruncaYngwebuildthenewbasisas:
11 ++ ⊗= lll sαα
AndtheHamiltonianforthenewblockas
...'0,01,1, +⊗+⊗+⊗=+ OOHIIHH lLllLlL
01, OIO llL ⊗= −with
..andfortherightblock
BeforetruncaYngwebuildthenewbasisas:
433 +++ ⊗= lll s ββ
AndtheHamiltonianforthenewblockas
...' 4,0)4(04,13, +⊗+⊗+⊗= ++−++ lRlLlRlR OOIHHIH
3+lβ4+lβ
3+ls
)1(0, +−⊗= lLlR IOOwith
PuzngeverythingtogethertobuildtheHamiltonian…
...' 3,1,
3,1,3,1,
+⊗+
⊗+⊗=
++
++++
lRlL
lRlLlRlL
OOHIIHH
1+lα 3+lβ
Trunca2onWhen we add a site to the left block we represent the new basis states as:
( )∑∑+
++
+
++
++++ ⊗=⊗=ll
lllll s
llslL
sllllll sUss
ααα
α
ααααα,
1,1
,1111
111
1
1+lαlα
1+ls
Similarly for the right block:
( )∑∑++
+++
++
+++
++++++ ⊗=⊗=43
34343 ,
43,3
,433433
lllll
ll slls
lR
sllllll sUss
βββ
β
βββββ
3+lβ4+lβ
3+ls
MeasuringobservablesSupposewehaveachainandwewanttomeasureacorrelaYonbetweensitesiandj
iO jO'
i j
WehavetwoopYons:1. MeasurethecorrelaYonbystoringthecompositeoperatorinablock2. Measurewhenthetwooperatorsareonseparateblocks
Weshallgoforop2on(2)forthemoment:simplerandmoreefficient
Operatorsonseparateblocks
iO jO'ˆ
i j
WeonlymeasurewhenwehavethefollowingsituaYon:
Then,itiseasytoseethat:
ββααψψ
αββαψψ
ψψ
βααβαββα
βααβαββα
ji
ji
jiji
OO
OO
OOOO
'''
'''
''
'',
*''
'',
*''
∑
∑
=
=
==
Wecannotdothisifthetwooperatorsareinthesameblock!!!
OperatorsonthesameblockiO jO'ˆ
i j
Weneedtopropagatetheproductoperatorintotheblock,thesamewayaswedofortheHamiltonian
Doneverdothis:
ααααψψ
ψψ
ααβαββα ji
jiji
OO
OOOO
'''
''
',
*'∑=
≠=
Targe2ngstatesinDMRG
If we target the ground state only, we cannot expect to have a good representation of excited states (dynamics).
If the error is strictly controlled by the DMRG truncation error, we say that the algorithm is “quasiexact”. Non quasiexact algorithms seem to be the source of almost all DMRG “mistakes”. For instance, the infinite system algorithm applied to finite systems is not quasiexact.
Our DMRG basis is only guaranteed to represent targeted states, and those only after enough sweeps!
Excitedstates
gsgsHHH Λ+=→ '
b)AteachstepoftheDMRGsweep,targetthegroundstate,andthegroundstateofthemodifiedHamiltonian:
FortargeYngthetwostates,weusethedensitymatrix:
1121
21
+= gsgsρ
a)Ifweusequantumnumbers,wecancalculatethegroundstatesindifferentsectors,forinstanceS=0,andS=1,toobtainthespingap
2DGeneraliza2on
WhydoestheDMRGwork???ωα
α
good!
bad!
Inotherwords:whatmakesthedensitymatrixeigenvaluesbehavesonicely?
EntanglementWesaythatatwoquantumsystemsAandBare“entangled”whenwecannotdescribethewavefuncYonasaproductstateofawavefuncYonforsystemA,andawavefuncYonforasystemBForinstance,letusassumewehavetwospins,andwriteastatesuchas:
|ψ〉 =|↑↓〉 + |↓↑〉 + |↑↑〉 + |↓↓〉
Wecanreadilyseethatthisisequivalentto:
|ψ〉 =(|↑〉+|↓〉)⊗(|↑〉+|↓〉)=|↑〉x ⊗ |↓〉x ->Thetwospinsarenotentangled!Thetwosubsystemscarryinforma2on
independentlyInstead,thisstate: |ψ〉 =|↑↓〉 + |↓↑〉
is“maximallyentangled”.ThestateofsubsystemAhasALLthe
informa2onaboutthestateofsubsystemB
TheSchmidtdecomposiYon
Universe
system
|i〉environment
| j〉
∑=ij
BAijABjiψψ
WeassumethebasisfortheleusubsystemhasdimensiondimA,andtheright,dimB.ThatmeansthatwehavedimAxdimBcoefficients.WegobacktotheoriginalDMRGpremise:Canwesimplifythisstatebychangingtoanewbasis?(whatdowemeanwith“simplifying”,anyway?)
TheSchmidtdecomposiYonWehaveseenthatthroughaSVDdecomposiYon,wecanrewirethestateas:
∑=r
BAABα
α ααλψ
Where
lorthonorma are ; and 0 );dim,min(dimBABAr ααλα ≥=
NoYcethatiftheSchmidtrankr=1,thenthewave-funcYonreducestoaproductstate,andwehave“disentangled”thetwosubsystems.
AuertheSchmidtdecomposiYon,thereduceddensitymatricesforthetwosubsystemsread:
∑=r
BABABAα
α ααλρ//
2/
TheSchmidtdecomposiYon,entanglementandDMRG
ItisclearthattheefficiencyofDMRGwillbedeterminedbythespectrumofthedensitymatrices(the“entanglementspectrum”),whicharerelatedtotheSchmidtcoefficients:• Ifthecoefficientsdecayveryfast(exponenYally,forinstance),thenweintroduceveryli3leerrorbydiscardingthesmallerones.
• Fewcoefficientsmeanlessentanglement.Intheextremecaseofasinglenon-zerocoefficient,thewavefuncYonisaproductstateanditcompletelydisentangled.
• NRGminimizestheenergy…DMRGconcentratesentanglementinafewstates.Thetrickistodisentanglethequantummanybodystate!
QuanYfyingentanglementIngeneral,wewritethestateofabiparYtesystemas:
∑=ij
ij jiψψ
Wesawpreviouslythatwecanpickandorthonormalbasisfor“leu”and“right”systemssuchthat
∑=α
α ααλψ RL
Wedefinethe“vonNeumannentanglemententropy”as:
22 log αα
α λλ∑−=SOr,intermsofthereduceddensitymatrix:
( )LLLLL S ρρααλρα
α logTr2 −=→=∑
EntanglemententropyLetusgobacktothestate:
|ψ〉 =|↑↓〉 + |↓↑〉
⎟⎟⎠
⎞⎜⎜⎝
⎛=
2/1002/1
Lρ
Weobtainthereduceddensitymatrixforthefirstspin,bytracingoverthesecondspin(andauernormalizing):
Wesaythatthestateis“maximallyentangled”whenthereduceddensitymatrixispropor2onaltotheiden2ty.
2log21log
21
21log
21
=−−=S
Entanglemententropy• Ifthestateisaproductstate:
{ } 0,...0,0,1 =→=→= SwRL αααψ
• Ifthestatemaximallyentangled,allthewαareequal
{ } DSDDDw log,...1,1,1 =→=→ α
whereDis
{ }RL HHD dim,dimmin=
Arealaw:IntuiYvepictureConsideravalencebondsolidin2D
singlet
2logcut) bonds of(#2log LS ≈×=
TheentanglemententropyisproporYonaltotheareaoftheboundaryseparaYngbothregions.Thisistheprototypicalbehavioringappedsystems.NoYcethatthisimpliesthattheentropyin1DisindependentofthesizeoftheparYYon
CriYcalsystemsin1Dcisthe“centralcharge”ofthesystem,ameasureofthenumberofgaplessmodes
EntropyandDMRGThenumberofstatesthatweneedtokeepisrelatedtotheentanglemententropy:
Sm exp≈
• Gappedsystemin1D:m=const.• CriYcalsystemin1D:m=Lα• Gappedsystemin2D:m=exp(L)• In2Dingeneral,mostsystemsobeythearealaw(notfreefermions,or
fermionicsystemswitha1DFermisurface,forinstance)…• PeriodicboundarycondiYonsin1D:twicethearea->m2
Thewave-func2ontransforma2onBefore the transformation, the superblock state is written as:
∑+++
++++++ ⊗⊗⊗=321 ,,,
321321llll ss
llllllll ssssβα
βαψβαψ
lα
1+ls
After the transformation, we add a site to the left block, and we “spit out” one from the right block
∑ ≈l
llα
αα 1
3+lβ
2+ls
∑++++
++++++++ ⊗⊗⊗=4321 ,,,
43214321llll ss
llllllll ssssβα
βαψβαψ
After some algebra, and assuming , one readily obtains:
∑++
++++++++++++ ≈31 ,,
343321114321lll s
llllllllllllll ssssssβα
ββψβαααψβα
Whathaveweleuout?
…ExploiYngquantumnumbers