Finite-temperature real-time field theories for spin 1/2

PHYSICAL REVIEW 8 VOLUME 39, NUMBER 10 1 APRIL 1989

Finite-temperature real-time field theories for spin —'2

Vitor Rocha VieiraCentro de Fisica da hfateria Condensada, Avenida do Professor Gama Pinto 2, 1699Lisboa Codex, Portugal

(Received 28 November 1988)

The finite-temperature real-time field theories for spin 2are derived with use of a functional-

integral approach, and their application to problems of equilibrium and nonequilibrium statisticalmechanics discussed. These formalisms have a standard W'ick's theorem and linked-graph theorem,leading to the use of Feynman diagrams. Similar to the problems with bosons or fermions, the cal-culation of equilibrium quantities requires the consideration of the imaginary-time branch, leadingto a three-component formalism, instead of the usual two-component formalism, typical of real-timefield theories.

I. INTRODUCTION

In recent years, there has been considerable interest inthe development and use of methods for the direct evalu-ation of real-time correlation functions at finite tempera-ture. ' Motivation for such interest comes fromdifferent fields. In condensed-matter physics, thosemethods are necessary when considering systems awayfrom thermodynamic equilibrium, critical dynamics,and in the study of disordered systems, ' for example. Inmathematical physics, they are useful in the study of sto-chastic processes and their applications, ' ' for exam-ple. In particle physics, ' they have been used in thestudy of the e6'ective potential at finite temperatures, '

in the study of spontaneous symmetry breaking in unifiedfield theories for the evolution of the early universe, andin situations with densities where the concept of tempera-ture becomes meaningful, as in a neutron star or in someplasmas produced with accelerators.

The study of equilibrium properties, or of the responsefunctions when the system is slightly perturbed fromthermodynamic equilibrium, which, using linear responsetheory, are expressed by thermodynamic averages, can bedone using the finite-temperature imaginary-time formal-ism'" first and then making an analytical continuation toreal time. ' ' However, when the deviation from equilib-rium cannot be considered as small, or in the formulationof a prob1em with an explicitly time-dependent Hamil-tonian, the imaginary-time formalism cannot be applied.Another situation of great importance is the study ofdisordered systems, where a disorder average has to betaken, in addition to the thermal average. Since the par-tition function, which is the normalization factor for thedensity matrix, depends on the disorder, one must dealwith the so-called denominator problem when doing thedisorder average. To solve this problem several methods,such as the replica trick ' ' or the superfield method,have been developed. However, the most physical ap-proach is to let the system evolve according to the disor-der Hamiltonian, for a sufficiently long time to reachthermodynamic equilibrium, starting from some arbitrarystate independent of the disorder. This is the basic idea

of using dynamics as a substitute for the replicatrick' ' ' in the study of disordered systems. In thisway, we are led to the consideration of a dynamical,nonequilibrium situation where the finite-temperaturereal-time formalisms are a natural tool. To ensure thatthe system approaches thermal equilibrium one has to in-troduce a relaxation and Auctuation mechanism whichcan be introduced phenomenologically using a Langevinequation approach or, more physica11y, coupling our sys-tem to a heat bath. '

For classical systems, a similar development of real-time techniques has occurred with the Martin, Siggia,and Rose formalism, and in this case the use of dynam-ics as a substitute for the replica trick also proved to beuseful, as in the spin-glass problem, ' for example.Those real-time techniques can be applied in general toany physical system consisting of bosons, fermions' oreven spins. However, the development of a useful per-turbation theory is normal1y based on a Wick'stheorem, ' resulting from the fact that, for boson or fer-mion operators, their commutators or anticommutators,respectively, are c numbers. Quite recently, the authorhas presented a finite-temperature imaginary-timefield-theoretic method for spin —,', with a standard Wick'stheorem. For spin —„ this is possible, since the spinoperators anticommute. For a general spin 5, the com-mutators of the spin operators are not c numbers, and asimple %'ick's theorem does not exist.

In the present paper, the finite-temperature real-timefield-theoretic methods for spin —,

' are developed, andtheir application to the type of situations consideredabove is discussed. In Sec. II, we formulate those situa-tions (nonequilibrium statistical mechanics, use of dy-namics as a substitute for the replica trick in disorderedsystems, and coupling of the system of interest to a heatbath to assure relaxation to thermodynamic equilibrium)in a unified manner, showing the appropriate generatingfunctionals to be used. In Sec. III, the functional in-tegrals for the evolution operator and the density matrixare obtained, using a representation of quantum mechan-ics or field theory by functions defined in phase space.The equation of motion for the trajectories and their ap-

39 7174 1989 The American Physical Society

39 FINITE-TEMPERATURE REAL-TIME FIELD THEORIES FOR. . . 7175

propriate boundary conditions are then derived. Togeth-er with the rule for the trace of an operator they lead tothe antiperiodic boundary condition of the imaginary-time formalism for spin —,'. ' In Sec. IV, we take atime-independent magnetic field as unperturbed Hamil-tonian and derive the generating functionals correspond-ing to the evolution operator and the partition function.The latter one is the generating functional for theimaginary-time formalism. ' Using the former wederive then, in Sec. V, the generating functionals for thereal-time formalisms, along with the propagators and di-agrammatic rules for the closed-time-path Green's func-tion (CTPGF) and thermo-field-dynamics (TFD) formal-isms. As it happens also in the case of problems involv-ing bosons or fermions, ' the treatment of equilibriumproblems and the calculation of quantities defined in ther-modynamic equilibrium require the consideration of theimaginary-time branch, leading to a three-component for-malism, instead of the usual two-component formalismcharacteristic of the real-time formalisms. Thiscorrects ' the original derivation of the thermo-field-dynamics formalism, as recognized independently bytheir authors. In the following paper, it is shown, inthe context of the Heisenberg model that the considera-tion of the imaginary-time branch is indeed necessary. InSec. VI, we list the different magnetization and suscepti-bility components since these are the quantities neededfor most applications, namely, those using the mean-fieldand the random-phase approximations. The simplest ofthem is the Heisenberg model, which is analyzed in thefollowing paper. Finally, in Sec. VII, we list our con-clusions. Some technical points and some details of thecalculations are presented in the appendixes.

II. REAL-TIME FIELD THEORIESAND THEIR APPLICATIONS

Practically every problem in statistical mechanics or infield theory can be reduced to the evaluation of somecorrelation function involving the expectation value ofthe product of several time-dependent operators Ak(tk),k =1, . . . , n, with respect to some probability distribu-tion given by a density matrix. The time evolution of theoperators is described by

(2.1)

(2.3)

having introduced, for convenience, a time t& larger thanall the times tk of interest. If the times tI, are chronologi-cally ordered, i.e., if tk+, ) tk, for k =1, . . . , n —1, thegenerating functional for this time-ordered product is

given by the product U2(t, , t&)U, (t&, t;) of two time-ordered exponentials. One is the forward time evolutionoperator

U, (t&, t; ) = T,exp i f —&'(t)dt (2.4)t

where the Hamiltonian &'(t) contains suitable time-dependent source fields associated with the different typesof operators, and the other is the backward time evolu-tion operator

Oz(t, , tI)=T, exp —i f gf(t)dt (2.5)L f

where, for the moment, it is not necessary to introduceany source terms. Taking functional derivatives withrespect to those sources, and setting them equal to zero atthe end, one immediately recovers the correlation func-tion of Eq. (2.3).

Due to questions of convergence, it is convenient toadd a small negative imaginary part to the infinitesimaltime differences dt in Eqs. (2.4) and (2.5). Those time evo-lution operators can then be interpreted as being definedon the oriented paths C& and C2, in the complex timeplane, respectively, as shown in Fig. 1.

If one introduces additional source terms in the time-ordered exponential U2(t;, t&), independent of those al-ready introduced in 0, (tI, t, )one is then .able to generateother types of correlation functions, with mixed time or-derings. An extreme case would be the correlation func-tions obtained using only sources in the path Cz, whichare antichronologically ordered. However, all thesecorrelation functions will be, of course, time orderedalong the combined oriented path C& UCz.

In most applications in statistical mechanics, one is in-terested in the grand canonical ensemble averages, with a

where, for convenience, t; is some time smaller than allthe times tk, k = 1, . . . , n of interest. The evolutionoperator U(t", t') is given in general by the time-orderedexponential

fl

U(t", t') = T,exp i f &—(t)dt (2.2)c,

reducing, for a time-independent Hamiltonian, to thesimple exponential e '~"" ", having used a system ofunits where 6=1. Due to the time ordering of the evolu-tion operator, the time-ordered correlation functions (orthe ones which can be interpreted as such) play a specialrole in the development of field-theoretic methods.

The product of the time-dependent operators is givenby

C

C3

FIT&. 1. Path C for the CTPGF formalism.

7176 VITOR ROCHA VIEIRA 39

density matrix given by

(2.6)

where /3=1/k~T is the inverse temperature, p is thechemical potential, and Z =Tr e ~'~ &+' is the partitionfunction. By the formal replacement i (tI —t;)—+P theequilibrium density matrix can be understood as theanalytical continuation of the evolution operator to imag-inary time, corresponding to the path C3 in the complextime plane, as shown in Fig. 1.

The generating functional for the finite temperature,real-time correlation functions is then given by the traceof the time-ordered exponential of the time evolutionoperator along the oriented path C =C, UC2UC3 in thecomplex time plane, as shown in Fig. 1. This leads to theso-called closed-time-path Green's-function formalism(CTPGF) developed by several authors. '

In the study of quenched disordered systems, a disor-der average has to be taken besides the thermal averagewhich we have just discussed. If one uses a density ma-trix of the type of Eq. (2.6), where the Hamiltonian & de-pends on the disorder, one faces the so-called "denomina-tor problem, " since the partition function Z depends onthe disorder, making it difficult to perform the disorderaverage. It is then preferable to use a different approach,starting at some time t =I;„ from some density matrixp( t, ), independent of the disorder and then letting the sys-tem evolve for a long time T =t& —t, ~ ~, until it relaxesto equilibrium. Since the time evolution of the densitymatrix is given by

(2.7)

the calculation of the expectation value of some operatorleads immediately to a picture of the type described byFig. 1, with the disorder coming in the paths C& and C2,but not in C3. En this manner, ' dynamics can be used ina physical way to solve the denominator problem occur-ring in the study of quenched disordered systems, and toavoid the use of some formal methods like the replicatrick2 ' ' or the superfield method.

In order to be sure that the system relaxes to equilibri-um it is necessary to introduce a fluctuation and relaxa-tion mechanism. The easiest way to introduce it and toderive a quantum Langevin or Fokker-Planck equation isto couple our system to a heat bath. One starts by writ-ing the generating functional for the system in the pres-ence of the heat bath. Since we are not interested incorrelation functions involving heat bath operators, onetakes the trace over the heat bath. The effect of the heatbath on our system is then given by an inAuence function-al comparable to the inhuence functional of the Feynmanand Vernon formalism. '

Finally, the so-called thermo-field-dynamics formal-ism ' ' (TFD) originally derived in an independentmanner from the closed-time-path Green's-function for-malism was interpreted as being equivalent to the use ofthe path C' in the complex time plane, shown in Fig. 2(with o. =P/2) instead of the path C of Fig. 1.

All these formalisms can, in principle, be used for any

FIG. 2. Path C' for the TFD formalism.

quantum system consisting of bosons, fermions, ' oreven spins. However, in the applications, it is impor-tant to have a practical perturbation theory scheme, likethe ones existing for boson and fermion operators, '

where standard Wick's theorems exist due to the fact thattheir commutators or anticommutators are c numbers,respectively.

For spin operators, in general, a simple Wick'stheorem does not exist, since the commutator of two spinoperators is another spin operator. However, for spin —,',the spin operators anticommute, and it is then possible tohave imaginary- and real-time formalisms similar tothose existing for bosons and fermions. We shall nowpresent the real-time formalisms using a functional in-tegral approach, since this is an extremely useful tool inthe formulation and in the development of approxima-tions for the solution of problems in statistical mechanicsand field theory. We start by considering a single spin,and later we make the extension to a lattice (or even acontinuum).

III. PATH INTEGRAL FOR SPIN 2

with I, m =x,y, z. In a path integral formalism, quantumoperators are represented by functional integrals overclassical variables. In the case of spin —,', the spin opera-tors anticommute, and it is then natural to useGrassmann variables as classical variables. They arecharacterized by their anticommutators

Ig', g I=0 . (3.2)

In this section we will obtain the path integral for spinWe will leave the details of the calculation for Appen-

dixes A, B, C, and D, stressing the key ideas in the maintext so that a reader familiar with Feynman's path in-tegral and with Grassmann variables, which are definedin Appendix A, will find natural Eq. (3.9) below, for thediscrete approximation for the functional integral, andEqs. (3.10) and (3.11), for its continuum-time limit.

For convenience of normalization, we define the opera-tors g'=o '/&2, where o ' is some representation forspin —„like the Pauli matrices, having anticommutatorsgiven by

(3.1)


As discussed in Appendix A, most of the usual opera-tions valid for real or complex numbers, -like differ-entiation, integration, etc., can be extended to the case ofGrassmann variables. Their anticommutativity automat-ically keeps track of the minus signs typical of fieldtheories with fermions.

In order to proceed, we must specify some definiteprescription establishing a correspondence betweenquantum-mechanical operators and functions defined inphase space. Such a prescription can be defined bychoosing some type of ordering for the operators and re-placing them by the classical variables. In the case ofspin —,', since the operators anticommute, we will use theantisymmetric or Weyl ordering. Given some functionof the spin operators, we expand it in powers, use the an-ticommutation relations of Eq. (3.1) to obtain a complete-ly antisymmetrized expression, and then replace theoperators g by the classical variables g . This correspon-dence, necessary to the phase-space representation ofquantum mechanics or field theory, and explicit ways ofestablishing it, is discussed in Appendix B. Thiscorrespondence, however, is not unique, due to the factthat o "a ~& '=i, similarly to what happens in generalwith Clifford algebras with an odd number of genera-tors, as discussed in Appendix C. As a result, eachoperator can be written in two different manners (or inweighted linear combinations of them), one with an evennumber of operators g and the other with an odd num-ber. If one chooses a representation with a definite paritythen the correspondence between operators and functionsbecomes unique. We will use the even representation, inwhat follows. The spin operators are then represented by

S= ——/X' .I

(3.3}2

In Appendix C, we determine how the functionrepresenting the product of two or more operators can bedetermined from the functions representing the operatorsand how the trace of an operator can be obtained from

the function representing it. As shown by Eq. (C2) theproduct g =g2g, of two operators g, and g2 is represent-ed by

—2~01.kg+&2 0+0 Cl ~

(3.4)

(3.7)

and represented by—i A(t& )t

U(rk+1 k ) —e (3.8)

where &(tk) is the "classical" Hamiltonian, i.e., the func-tion in phase space representing it. In practice, givensome spin Hamiltonian one simply has to replace the spinoperators by their classical expression given by Eq. (3.3).Using Eq. (C10) of Appendix C for the product of aneven number N of operators, one arrives at the discreteapproximation for the evolution operator of a spin- —,

' sys-tern:

where dgk =dgdgdg for k =1,2, with integrationover Grassmann variables as defined in Appendix A.When using the even representation, the trace of anoperator is given by

Trg =2g(0) (3.5)

as one can see from Eq. (C19).Normally, in the development of a functional integral

expression for the evolution operators 6'(t&, t, ), one fac-tors it in the product

U(tfyr/) U(rfyr~, } . . U(tk+„tk) 0(t, , t; ) (3.6)

of N infinitesimal time-evolution operators, introducingintermediate times tk =t, +kit, for k =0, 1, . . . , N,where b, t =(tI t, )/N, a—nd . having defined to=t;, t~ =t&Keeping only terms up to order 1/N which is formallyjustified in the limit N~ ~, the infinitesimal evolutionoperator is then approximated by

U(tk+ ), tk ) = 1

iaaf'(tk

)b,t—

01

K(g, tI, t;)= f f exp (g, . g~ g) —11

23%/2

0 —1 1 —1

1 0

—1 1 —1 0N

N —1

i g &(tk—)b, t dg, . dg~ .k=0

(3.9)

K(g, r&, r, )= fg)ge's, (3.10)

This can be taken as the starting point for the develop-ment of a lattice theory having ht as the lattice constantin the time direction. Taking the continuum-time limit,obtained letting N —+ ~, one obtains, as discussed in Ap-pendix D, the functional integral for spin —,, given by

where the action S is given by

+g(t; ).g+ g g(tI ) . (3.11)

One recognizes the presence of three terms. The first is

is= —f —g- +i%(t) dr+ 'g(t&) g(t,)-1 dg2 dt f

VITOR ROCHA VIEIRA 39

,'[g(t&)+—g(t,)] (3.13)

showing that the center of gravity of the end points of thetrajectories must be g, the point in phase space, whereK(g, t/, t, ), the function representing the evolution opera-tor U(t&, t, ) in phase space, is defined. The same equa-tions of motion and boundary conditions can be obtainedvarying the action for the discrete approximation for thefunctional integral and then taking the continuum limit.

All these results can be immediately applied to the den-sity matrix p=(1/Z)e, by the Wick's rotation inwhich it is replaced by ~. The Boltzmann factor e ~ isthen represented by

K(g, p)= f2)ge

where the "Euclidian" action is given by

(3.14)

S= f —'g +&(~) d ~——,' g(P) g(0)0, d

—g(0).g —g.g(g) .

The equations of motion are given by

(3.15)

similar to the pq term in Feynman's path integral, mak-ing the transition from the Hamiltonian to the Lagrang-ian. This term comes from the quantum nature of theproblem and is associated to the statistics of the fieldsused. The second term is the classical Hamiltonian asdiscussed above. Finally, the last term is given by theboundary terms appropriate for the antisymmetric orWeyl ordering which we are using. These boundaryterms are important for the boundary conditions associ-ated with the equations of motion for the classical trajec-tories or for the Dyson-Schwinger equations. Varyingthe action of Eq. (3.11) one finds the equation of motion

. dg B&(3.12)

dt Bg

where 8/Bg is the left derivative, obtained taking thedifferential dg to the left, and the boundary condition

imaginary-time formalism for spin —,'.In quantum mechanics, where one starts from the

Hamiltonian formulation, the difficulties in the derivationof the formalism come from the dependence on time, andthe consideration of other degrees of freedom such asmomentum or internal degrees of freedom poses, in gen-eral, no problem. In problems in statistical mechanicsone normally deals with a system composed of a largenumber of particles. The formalism that we havedeveloped can be immediately applied to a problem in-volving a large number of spins, simply using our resultsto each of the spins. In general, there will be interactionsamong the spins, which have to be taken into account us-ing either perturbative or nonperturbative methods.

IV. SPIN IN A TIME-INDEPENDENTMAGNETIC FIELD

Having in mind the development of perturbation-theory formalisms, we will now evaluate the functionrepresenting the evolution operator of a spin —,', in thepresence of a time-independent magnetic field and of atime-dependent source field p(t) coupled linearly to theelementary fields g, which we introduce in order to gen-erate correlation functions taking functional derivatives.The Hamiltonian is given by

Qo= —H S+ip(t).g . (4. 1)

Since S is quadratic in the fields g, we are led to aGaussian functional integral, which can be calculated ex-actly completing the square as given by Eq. (A9) of Ap-pendix A. An easier way is to evaluate the action for theclassical trajectory given by the classical equations ofmotion. One then has to evaluate the prefactor given bya functional determinant. It is a simple matter to verifythat this still holds for Gaussian integrals overGrassmann variables.

The equation of motion obtained from Eqs. (3.12) and(4.1) describes the precession around the magnetic field

H, according to

(3.16) dt=gXH —p . (4.2)

and the boundary condition by

0= —,' M(»+ 0(0) l . (3.17)

Z =2f2)g exp —f —,'g. +&(r) dc0 d1

(3.18)

where the trajectories are antiperiodic in the imaginarytime, i.e., they satisfy

g(P) = —g(0) . (3.19)

This expression and this antiperiodicity condition can beused for the development of the finite-temperature,

The partition function Z =Tr e ~ involves the evalu-ation of the trace. According to Eq. (3.5) and to theboundary condition of Eq. (3.17) the functional integralfor the partition function for spin —, is then

The fields S= i /2$X g als—o precess around the magnet-ic field, according to the equation

dSdt

=SXH+igXp . (4.3)

In order to solve the equation of motion (4.2) it is use-ful to introduce a reference frame with the third axisalong the direction of the magnetic field, and then to usea spherical basis defined by a+—=(1/&2)(e, +ie ),ao=e„associated with it. We would like to stress that the vec-tors a+ and the spin components which we will define inthe following, are normalized, i.e., they are defined withthe factor I /v'2, as opposed to the usual conventions. Inparticular, the dot product of two vectors S, and S2is Si S2=Si+S2 +Si S2 +SiSz =g SiSz, where a=+,—,0 and @7=—a. In this spherical basis the equa-tion of motion becomes

39 FINITE- TEMPERATURE REAL-TIME FIELD THEORIES FOR. . . 7179

1S

~ di =aHP i—pdt

(4.4)

The solution, with the boundary condition of Eq. (3.13)

It is given by

e(t —t ) e(t —«)

1 +e—iaHT

&laHT+ 1

p(t) = e—iaH(t —t)

P—i f G'(t, t')p (t')dt',I

(4.5)

aH G—'(t, t') =5(t —t'),ddt

(4.6)

where t=(tf+t;)/2, T=tf —t;, and G (t, t') is theGreen's function defined by

—i aH(t —t')Xe (4.S)

This solves the equation of motion. One now must evalu-ate the action of Eqs. (3.11) and (4.1) for this trajectory.Using the equation of motion (4.4) one has

tfiS'= g f ,'p, (t)—p(t)dt+,'p(tf-)g (t, )t

with the boundary condition+0 (t )P+PP(tf) (4.9)

G'(t, , t')+G'(tf, t')=0 . (4.7)Using the expression (4.5) for P(t), and those for P(t;)and P(tf) obtained from it, one finds that

tf e i aH( t —t ) iiS = g i tanh— g~P —f dt P p (t) —f—dt f dt'p (t)G (t —t')p (t')2 &; cos(aHT/2) 2

(4.10)

i aH 5@=—k,„g'-ddt

(4.1 1)

with the boundary conditions

5p(tf )+5@(t,)=0, (4.12)

The prefactor giving the contribution of the Gaussianffuctuations 5p around the classical trajectory can becalculated evaluating the functional determinant of thediff'erential operator id/dt aH of the qu—adratic part ofthe action. The eigenvalues A,„are defined by thedi6'erential equation for those fluctuations

T

K (g, tf, t; )=cos HT2

exp A+ g( —2',

iU(p)—

having used the infinite product formula for the cosine.The alternative is to use the discrete approximation forthe functional integral and finding a closed expression forthe finite product over n, before taking the continuumlimit X~~.

The function representing the evolution operator in thepresence of a time-independent magnetic field and of atime-dependent source p, (t) is then given by

since the classical trajectory already satisfies the bound-ary condition of Eq. (3.13). They are

with(4.16)

A,„—c'o„aHwhere

co =(2n +1)—.n T '

(4.13)

(4.14)

i tfA = ——' f dt f dt' g p, (t)G'(t t')p (t'),—

tf l'aH(t —t)dt p (t)

cos(aHT/2)

(4.17)

(4.18)

The functional determinant is proportional to g„A,„.Using as normalization factor the determinant for thecase of a zero Inagnetic field, the prefactor is thereforegiven by

and

aHTU =tan (4.19)

Det i —oHdt

Det idt

1/2

g (co„H)co„—Q co„co„

The prefactor and the last term of the action is whatone finds evaluating directly the function representing theevolution operator without sources, according to

e ' =cos

n)0HT=cos

21+i2 S tan

2

HT=cos2

(4.15)HT . 8icos exp i 2 .S tan

HT2

(4.20)


and replacing the operators S by their expression in termsof the elementary fields given by Eq. (3.3).

We can verify that our expression for K (g, t&, t, )

satisfies the equation of motion for the evolution operator

. dUi =&U, (4.21)

dt

which, in phase space, is represented by

Xe ' ' ' ' dg, diaz . (4.22)

The function representing the non-normalized densitymatrix T,exp[ —f~&&(r)dr] is simply obtained perform-

ing the Wick rotation it~r, it;~0, it&~P. One thenhas

K (g,P)=cosh H2

is the propagator

(4.28)

of the imaginary-time formalism, since, from Eq.(3.5) for the trace, the partition function in the presenceof sources v (r) is given by

Z[v] =2 cosh2

r

Xexp ,'—f—drf dr'0 0

X g v (r)Q (r r')v —(r')

(4.29)

This propagator is antiperiodic in imaginary time, andits Fourier transform is

X exp A + g (2ig v —V Pg )a

(4.23)

& (8„)= 7

I CO~ O.'H(4.30)

where

aH(, ~—Pj2)v —— dv v (~),

0 cosh(PaH /2)(4.25)

V =tanh aH (4.26)

,' f —d—rf dr'g v"(r)Q (r—r')v (r'), (4.24)0 0

where co„=(2n+1)vr/P, n =0,+1,+2, . . . are the fer-mionic Matsubara frequencies.

Our expressions for K (g, t&, t; ) and K (g, P) are one ofthe major results of this paper. They will be used to ob-tain the generating functionals and propagators for thereal-time formalisms and in the formulation of nonequili-brium approaches to study spin- —,

' systems far away fromthermodynamic equilibrium. We will now proceed to thederivation of the real-time field theories.

Finally, V. REAL-TIME FIELD THEORIES FOR SPIN~

e(r —r')1—paH+ 1

1 —aH( r—r')

1+ paH

(4.27)

As discussed in Sec. II, the closed-time-path Green'sfunction (CTPGF) and the thermo-field-dynamics (TFD)formalisms have as their generating functionals the traceof the evolution operator along the paths C and C' in thecomplex time plane, respectively. One is then led toconsider

L

Z[v, p', v', p]= —TrT exp —f &"(r)dr exp —i f &"(t)dt exp —f & (r)dr exp i f Vf i—'(t)dtZ . O' . . I

(5.1)

having introduced sources p, v', p', and v, respectively, in the branches C„C2, C3, and C4 of Fig. 2, respectively.The procedure now is to replace each of the evolution operators and density matrices of Eq. (5.1) by the functions

representing them in phase space and then use Eq. (C10) and (C19) to obtain their product and to take the trace. In Ap-pendix E we show these calculations in some detail. We shall discuss our results here.

Let us consider first the TFD and the CTPGF formalisms. When using them one considers correlation functions in-

volving operators with real-time arguments corresponding to the horizontal paths of Fig. 2. The original formulation ofTFD is obtained for o =/3/2, and the CTPGF for o. =0. One does not need to introduce source fields in the imaginary(or vertical) branches of the p'ath C' of Fig. 2. Taking v(r) =v'(r) =0, and using the results of Appendix E, one obtainsthe free generating functionals for the TFD and CTPGF formalisms. From Eqs. (E6) and (E7), they are given by


Z[p', p]=e xp f dt f dt'[p(t)(p') , (t')]G (t t—')i t

where the free propagator G (t t—') is given by

(5.2)

e(t —t )—PaH+ )

e(t —t)l+ep H

e—(P/2 —o )aH

hpaH

G '(t t') =— t—e (P/2 —o. )aH

PaH2 cos

6(t t')—) +ePaH

6(t' —t)—PaH+ )

—iaH(t —t')e (5.3)

For time-invariant systems it is useful to work in frequency space. Letting t,.~—~, t&~+ ~, introducing the con-vergence factors e " ' ' in the limits t —t'~+ ~ and defining the fields

p (co)= f dt e'"'p (t},with similar definitions for the fields (p ) (t), one obtains

(5.4}

Z[p'(co), p(co)]=exp f [p ( —co)(p') ( —co)]G (co)2 2&

where G (co), the Fourier transform of G (t t') i—s given by

(5.5)

6 (co) =

1 1 1 1

e ~ +1 a —oH+l'6 1+e~ H co —aH —i6

27Fl. e ( p/2 —0. )aH

6(m —aH)paH

2 cos

27Tl. e

—(P/2 —o )aH5(cu —aH)

hPaH

1 1 1 1

+ ePaH CO—aH+l5 e

—PaH+ 1 CO—CtH —l5

(5.6)

ifc aWe would like to stress that since p (co) =p (—co), p (co) is not independent of p (co). This is important when taking

functional derivatives, and is the reason for having introduced the factor —, in the exponents of Eqs. (5.2) and (5.5).This propagator has the standard form of the propagator in the TFD and CTPGF formalisms, as we can see making

the natural identification e =aH between the energies aH of the fields P and the single-particle energies e of thoseformalisms.

When cr=p/2, this propagator is quite symmetrical and can be diagonalized by a Bogoliubov-Valatin transforma-tion. In fact, defining the propagators

g„—(co)=CO CXH+l 5 (5.7)

given in real time by

g+(t t')= ie(t t')e' ~—"— —

g (t —t') =i6(t' —t)e

(5.8)

(5.9)

and defining

1 1cosOa = slnOa =

PaH+ 1 )1/2 a( 1+—ePaH)1/2

cose sine

one finds that, for cr =p/2, the propagator G (co) can be written as

cose sine g ( co ) 0

(5.10)

G (co)= —sine cose 0 —g (co) —sine. cose (5.11)


0 0 0 0G (co cr ) = 0 G (coP/2) 0 1/, (512)

with u =e') ~' . If one considers to be "physical"only those correlation functions involving fields from theforward or physical real path only, o. can be completelyeliminated by a nonunitary redefinition of the fields

(g') (co) and (g') (co)=(g'} ( —co), in the "unphysical"or backward path (corresponding to the annihilation andcreation operators of the usual field theories), given byg'~ u g' and (g')*~ 1/u (g')'. Since the physical correla-tion functions always have the fields g' and (g')' comingin pairs, they do not depend on o, as expected. 36

The propagator for the CTPGF formalism is obtainedtaking o.=0. Defining the Fermi functions

1

+PaH+ 1(5.13)

and using the identity

showing that the propagator at finite temperature can beobtained from the zero temperature propagator, havingthe propagators g —(co) completely decoupled, by a fer-mionic Bogoliubov-Valatin transformation. In the origi-nal derivation of TFD formalism, Takahashi andUmezawa proceeded in the opposite direction, i.e., theyexpressed the expectation value in the grand canonicalensemble as an expectation value in a thermal state ~P),which, in the case of free particles, is connected to thevacuum by a Bogoliubov-Valatin transformation. Theextension to a system of interacting particles was madeusing the quasiparticles associated to the interactionHamiltonian. The definition of the thermal state ~P) re-quires the duplication of the degrees of freedom of thetheory, i.e., it requires the introduction of the fields corre-sponding to the backward evolution operator, and onearrives at a two-component field theory.

The quasiparticles diagonalizing the Hamiltonian, asstressed by Takahashi and Umezawa, are defined by aself-consistent condition, which is therefore nonperturba-tive and temperature dependent. In particular, the quasi-particle energies are temperature dependent and do notcoincide with the bare energies appearing in the originalHamiltonian. We will discuss later in this section theneed to consider also fields from the imaginary branch,leading to the use of a three-component formalism,and how the Takahashi-Umezawa condition should be in-terpreted in nonperturbative calculations.

For the original formulation of TFD one has cT =P/2,corresponding to a symmetric factorization of the parti-tion function, ' in the construction of the thermal state~I3), as can also be seen from expression (5.3) or Fig. 2.For time-independent Hamiltonians the value of o. is ir-relevant. This can be seen relating the propagator ofEq. (5.6) for general cr with the propagator for (r =P/2,according to

g+ (co)—g (co)= 2—re 5(co a—H), (5.15}

the propagator can be written as

(f.g.'+f.'g. )0(~)= f (

+)a ga ga

f.'(g.' —g. )

—(f.'g.'+f.g. }

(5.16)

It has the remarkable property that the sum of all its ele-ments is zero. This reAects the normalization

Z[p'(t), p(t)]~ .(,) (,)=1 (5.17)

of the generating functional. For equal sourcesp'(t)=p(t), the Hamiltonians &" and &" in the tworeal branches are the same and, therefore, the two evolu-tion operators cancel each other. One then finds Eq.(5.17), since the density matrix is normalized to one.

More generally, decomposing G (t t') in—

G (t t')=G —(t —t')e(t t')+G —(t t')e(t' —t)—(5.18)

it is easy to verify that one must have

(11)G o (t —t')=0 (5.19)

and

G' (t t')1—=0. (5.20)

These normalization conditions can be stressed,defining the fields P= —,'(g+g') and g"=g f, which—canbe considered as the center of gravity and the relativecoordinate of the fields in the two real branches. In thisbasis the propagator takes the form

G (~)=0

,'(f.+ f. }(g.'———g.)— (5.21)

0 1

co aH 15

1 . PaHim tanh 5(co aH)—

co —aH+i 5 2

(5.14)

to write

1 1=P +i~5(o) aH)—co —aH+i 5 co —aH

(5.22)

having a zero in one of the components, separating clear-ly the dependence on temperature and the advanced and


retarded Green's functions.As we have said before, a system with very many spins

—,' is treated simply applying this formalism to each of theindividual spins. We have derived the free propagatorcorresponding to the situation where the spins are in atime-independent magnetic field, not necessarily spaciallyuniform. This magnetic field does not need to be theexternal applied magnetic field, but can be the molecularfield of a mean-field or self-consistent Hartree calcula-tion. A nontrivial problem normally requires the use ofperturbation theory techniques. One then separates thetotal Hamiltonian in the sum of two parts: one, &o, theunperturbed Hamiltonian, in general quadratic, leadingto a propagator of the type we have just obtained, andanother, &;„,=&—&o, to be handled by perturbationtheory. Using the functional integral approach which wehave developed, one makes an expansion in powers of&;„„uses Wick's theorem to obtain the contractions andto arrive at the Feynman's diagrams. From the explicitform of the interaction Hamiltonian one then derives thediagrammatic rules to be used. We will only com-ment that the fields g ( —co) and g (co) are not indepen-dent and that, therefore, in the expectation value of 2nfields there are (2n —1)!!contractions. The frequency co

which we have introduced is convenient to take care offrequency conservation at each vertex for a time indepen-dent Hamiltonian, but the arrow direction is just a matterof convention. Diagrams can be associated in pairs rev-ersing the position of the fields g in each spin S, cancel-ling the factor —, in its definition. Closed loops contributewith a relative minus sign.

From expression (5.1), one sees immediately that theinteraction Hamiltonian appears in both the forward andbackward evolution operators. Since the propagator Ghas nonvanishing elements G i2 and G 2„connecting thefields in the two real branches, one realizes immediatelythe need for the characteristic duplication of the degreesof freedom in the real-time field theories. Besides thefields representing the system, also needed is a "copy" ofthe system. ' From expression (5.1) one also sees thatthe Lagrangian for the system plus its copy is thedifference of the individual Lagrangians for the systemand its copy. ' The application of the real-time formal-isms for spin —,, which we have developed, is similar tothe case of bosons and fermions, and most of the con-siderations usually made, concerning perturbative andnonperturbative methods, renormalization-group tech-niques, ' etc. , can be immediately applied.

As already stated in Sec. II, correlation functions in-volving fields from the backward branch are also physi-cal. Simply they are not time ordered. They will be ad-vanced or retarded according to the time orderingdefined by the path C or C' in the complex time plane.

An important point we would like to discuss now is therelevance of the imaginary branch. In the argumentwhich we have presented to justify the duplication of thedegrees of freedom in the real-time theories, we left outthe fact that the perturbation Hamiltonian is alamo con-tained in the unnormalized density matrix e ~ . Oneshould also look for propagators of the type G &3, G 23 and

G 3i, G 3p, connecting the fields in the real and imaginarybranches. It is sometimes argued that, due to factors of

+lCOt + l COtIthe type e ' or e appearing in those propagators,which are rapidly oscillating in the limit t; ~—~,tf + + ~ the imaginary branch decouples from the oth-er two. ' This is actually not true and affects, in particu-lar, the calculation of quantities defined in thermodynam-ic equilibrium. The time evolution is correctly given,since &;„, appearing in the evolution operators is takeninto account, and one is still able to find the correctionsto the self-energies, for example. However, the thermalaverage is replaced by the thermal average with respectto the unperturbed Hamiltonian &0, and, therefore,quantities like occupation numbers are affected. In orderto find the corrections to them, we have to consider theimaginary-time branch, arriving at a field theory withthree fields for each degree of freedom, ' ' instead ofthe usual duplication of degrees of freedom, characteris-tic of the real-time theories. In the following paper,where this formalisms are applied to the Heisenbergmodel, these features are found and the need to considerthe imaginary. -time branch verified in a specific problem.

In the treatment of a nonequilibrium problem, howev-er, the use of the usual two component formalism is stilljustified, because neglecting the imaginary-time branchonly affects the initial density matrix, and initialboundary conditions normally are not relevant, since thesystem relaxes to thermal equilibrium eventually, i.e., inthe limit t, ~—ao. However, in order for this to happen,it is, in principle, necessary to have a fluctuation and dis-sipation mechanism which can be introduced couplingour system to a heat bath. Its effect can be expressed byan inhuence functional comparable to the inhuence func-tional of the Feynman and Vernon formalism. ' ' Thispoint will be further developed in future publications.

In the original derivation of the TFD formalism,Takahashi and Umezawa stressed that their perturbationtheory was not a simple perturbation expansion in theoriginal coupling constant. Their approach can be inter-preted as a self-consistent resummation of diagrams, lead-ing to new quasiparticle energies, and eliminating classesof diagrams where the imaginary branch appeared. How-ever, a nonperturbative treatment, even using these quasi-particle energies, would eventually break down and leadto inconsistencies in the treatment of an equilibriumproblem. '

We now present the remaining propagators for thethree-component formalism. We will consider the exten-sion of the CTPGF formalism, for which cr =0, since it isthe most interesting case. Going back to the beginning ofthis section, and keeping the source v(r), one finds, usingagain the results of Appendix E, new terms for our gen-erating functional Z [v,p', p], besides those of the real-time formalism, given by Eqs. (5.3) and (5.6), which nowconstitute the real-real part of the three-component for-malism. The imaginary-imaginary part is given by theimaginary-time formalism, as in Eqs. (4.27) and (4.30).Finally, from Eq. (E8) the mixed-time components aregiven by


1exp i—f dt f dr+ [p (t)(p') (t)jg (t, ~)

1v (r)

a

p ~y p (t)=exp —i f dr f dt g v (1—1)g' (r, t)

o t;' (p') (t) (5.23)

where

] —iaH(t —t,. )

g (tr)= e 'ea ' 1+ paH (5.24)

and

1 i aK(, t —t,. )g' (r, t)= — e 'e—Paa+&

(5.25)

A technical remark is in order at this point. The Fourier transformation is useful for time-invariant systems, becauseit transforms convolution equations into algebraic ones. The reason for this is that a function of the type g(t t ),—which strictly speaking should be considered as a matrix with entries t and t, is diagonalized by the Fourier transfor-mation. In principle, one should make independent Fourier transformations in each of the variables according to

f dt e' 'f dt'e ' 'g(t t') =g(co)2—~5(co —co') (5.26)

to obtain a diagonal matrix in Fourier space. In practice, however, one drops the delta function and makes a singleFourier transformation in the time difference. The same applies to the imaginary-time formalism where one would findthe Kronecker delta function 135 „instead of the Dirac delta function 2@5(to—co'). For the mixed terms, although oneis still dealing with a difference of two time arguments, one is forced to make separate Fourier transformations. Theyhave different characters, one being an integral over a continuum of frequencies, and the other a series over discrete fer-mionic Matsubara frequencies. As a result, we have to make also double Fourier transformations in the purely real andpurely imaginary parts of the formalism, leading to the use of the delta functions referred to above. In frequency space,the propagator for the three-component formalism is therefore given by

«.g.'+f.+g. ) f+(g+ —g )

G (co, co') =—ig g e '(1 —1) g )335

(5.27)

where, for convenience, we have introduced the notation

g (~)=LCO AH

(5.28)

We would like to stress that the formalism which wehave presented, using a functional integral approach,and, in particular, the techniques contained in the appen-dixes, can handle very general situations, providing thenatural tool for the treatment of nonequilibrium spin- —,

'

systems. The application to the study of disordered sys-tems and the comparison to the recent developments ofthe nonequilibrium thermo-field-dynamics will be madein future publications.

In the following paper, this formalism will be appliedto the study of the Heisenberg model, showing the needto consider the imaginary branch, leading to the three-component formalism.

magnetization and susceptibility components, which arethe lowest-order spin-correlation functions. As a matterof fact, most approximations used in the literature, name-ly the mean-field approximation and the random-phaseapproximation require only the knowledge of these corre-lation functions. Treatments beyond these two approxi-mations will require the use of higher correlation func-tions, which can be calculated using again the formalismof the preceding section.

The magnetization is defined as the expectation valueof the spin operator

S,(t)= ——,' pe, g (t)P(t)

a, P

in the spherical basis, where e & is the antisymmetricsymbol, with eo+ =1, as follows from Eq. (3.3). Thespherical spin components are diagrammatically shownin Fig. 3. The magnetization is then given by

VI. LOWEST-ORDER CORRELATION FUNCTIONS:MA(GlNETIZATION AND SUSCEPTIBILITY M;(t) = —g e, ,G "(t=0 )

a(6.2)

Having derived, in the previous section, the real-timefield-theoretic formalisms for spin —„we will now list the

showing that the only nonvanishing spherical componentis for y =0, meaning that the magnetization points in the


direction of the magnetic field, as expected. Using Eq.(5.3) for the propagators of the elementary fields g, onefinds

Mo(t) =—,' tanh (6.3)

independent of time and branch.The next correlation function to be considered is the

spin-spin time-ordered cumulant

g'tt(t —t') = ( T&'(t)&z(t') ), . (6.4) I~

From Eq. (6.1) one finds FIG. 3. Spherical spin components.

y'b&(t t')=—,'e „,e—& ~G0 (t t')G ib'„—(t' t)—

,'5 tt[G-00 (t t')G—' (t' t)+G—' (t t')G,'(t'—t))—+ ,'6 &

g-G'„(t t')G„'—(t' t) G—' (—t —t')Go& (t' —t)P

Gop (t t')—G' —(t' t) G',—(t ——t')G', (t' —t) (6.5)

The susceptibility, defined as in Eq. (6.4), is diagonal in spin space, as expected. The transversal susceptibilities aregiven by

y"(t t')= ,'[G', (t—t')G—' (t' t)—+G' —(t t')G,' (t'——t)] (6.6)

for a=+, as could be obtained directly from Fig. 3, for the spherical spin components. We recall that we are using nor-malized spin components. Our susceptibilities g+ and g correspond to —,'g+ and —,'g +, respectively, in the usualnotation.

Using again the propagators of Eq. (5.3), one finds that the transversal susceptibilities a =+ have

y'"(t t') =aM—e

—craH

—PaH

eo.aH

e PaH—i aH(t —t')

e(t t')+ — e(t' t)—1, 1

PaH 1 —e

for a, b = 1,2, as real-time components (having kept o &0),

(6.7)

(r—r') =aM 1, 1—PaH PaHe(~—r')+ e(r' —r) e (6.8)

as an imaginary-time component, and finally

1 —iaH{t —t,. )y' (t, r)=aM e ' ee PaH

M =—' tanh2 2

was used. The longitudinal susceptibility is given by

(6.10)

and yo (t —t') = —,' g G „' (t t')G „' (t' t) .— —(6.11)

1 iaH(t —t,. )

e—PaH (6.9) The real-time components are given by

for a =1,2 as mixed-time components, where the nota-tion

1 1

yo (t —t') = (6.12)


the imaginary-time component by

yo (r —r') = (6.13)

and the mixed-time components by

XoXo («)=So (& r) = (6.14)

where

1Xo 4 cosh (pH/2)

(6.15)

is the usual longitudinal susceptibility.For time-invariant systems it is convenient to go into

frequency space. Defining the Bose functions

+= 1II

e +PaH (6.16)

and using the notation of Eqs. (5.8), (5.9), (5.15), and(5.28), the transversal susceptibility has real-time com-ponents given by

(n g+' —+n+g. ) n+(g+ —g )

—n (g+ —g )y' (cu) =iaM- (n+g+ +n g )

(6.17)

or this expression multiplied by 2~5(co —co ) if one is us-ing the three-component formalism; the imaginary-timecomponent given by

(co)= —aMg (co) (6.18)

Similarly, for the 1ongitudinal susceptibilities one finds

1 1

yo (~)=1 1

2vro(co), (6.20)

or this expression multiplied by p6, ; and mixed-time

components given by

y' (co,co)= iaMg+—(co)g (co)e(6.19)

y '(co, co)=iaMg (ci))g (co)e

We started with the derivation of the functional in-tegral expressions for the evolution operator andBoltzmann factor and discussed the appropriate bound-ary conditions to be used in connection with the equa-tions of motion. When applied to the partition functionthey lead directly to the antiperiodic boundary condi-tions, in the imaginary-time direction, which had beenfound and used in the finite-temperature imaginary-timeformalism.

We then considered the case of a time-independentmagnetic field with a source term added, and evaluatedexplicitly those functional integrals, using the sphericalbasis associated with the magnetic field. They were thenused to obtain the generating functionals for the real-timeformalisms, giving the free propagators for those formal-isms. The perturbation theory rules for problems withnontrivial interactions were indicated. We then listed themagnetization and susceptibility components, which arethe correlation functions needed for most applications in-volving spins, namely those involving the mean-field andthe random-phase approximations.

The calculation of quantities defined in thermodynamicequilibrium requires the consideration of the imaginary-time branch, in order to have the thermal averagecorrectly given. This leads to a formalism with threefields for each degree of freedom, instead of the usual du-plication of degrees of freedom, characteristic of thereal-time theories. In the following paper, where this for-malism is applied to the Heisenberg mode1, the need forthe consideration of the imaginary-time branch, for cal-culations in a situation of thermal equilibrium, is verifiedin a specific problem.

The formalism which we have developed is of generaluse and provides, in particular, the natura1 tool for thestudy of nonequilibrium situations or to the study ofquenched disordered systems, using dynamics as a substi-tute for the replica trick. In order for the system to relaxto thermal equilibrium, a fluctuation and relaxationmechanism must exist, which can be introduced couplingthe system to a heat bath. The application of those tech-niques to this type of situation will be considered in fu-ture publications.

and

yo (co)= p5 (6.21)ACKNOWLEDGMENTS

I would like to thank H. G. Schuster and I. R. Pimen-tel for several stimulating discussions.

Loyo'(co, co) =go'(~o, co) = 2~5(co)P5 (6.22)

APPENDIX A: GRASSMANN VARIABLES:DEFINITIONS AND PROPERTIES

VII. CONCLUSIONS

We have derived the finite-temperature real-time field-theoretic formalisrns for spin —,', i.e., the so-called closed-time-path Green's-function, and thermo-field-dynamicsformalisms for spin —,', providing the extension to realtime of the finite-temperature imaginary-time formalism,which had been previously developed, and discussed theequilibrium and nonequilibrium applications of those for-malism s.

In this appendix we define Grassmann variables anddiscuss some of their properties. Grassmann variablesare characterized by their anticommutation relations

(A1)

As a result, the square of a Grassmann variable is zero,and the most general function of a finite number ofGrassmann variables is a polynomial with the highestmonomial having the product of all the variables.

Complex conjugation is the analog of Hermitian conju-

39 FINITE-TEMPERATURE REAL- TIME FIELD THEORIES FOR. . . 7187

gation. In particular, it satisfies

(g*)*=g (A2)

and reverses the order of the operators in a product, i.e.,

(pv)4 —v4p4 (A3)

Real variables do not change under complex conjugation,i.e., are those satisfying p* =p. Differentiation stillsatisfies the usual linearity properties. However, whentaking the differential of a product, one should notchange the order of the operators, i.e., one must write

d(pv) =(dp)v+p(d v), (A4)

ff((+A,,)dg= ff(g)dg . (A5)

similarly to what happens with the differential of thecross product of two vectors.

Left 8/(i(](, and right (l/B(M derivatives are defined ac-cording to which side the differential d(u is removed (leftor right). Integration also follows the usual linearityproperties and is invariant under a shift in the integrationvariable, i.e., one has

and is simply related to the determinant by[P(A)] =Det(A) as can be easily verified taking thesquare of Eq. (A9), for (L(=0, and changing to complex in-tegration variables.

Taking functional derivatives of Eq. (A9) it is straight-forward to obtain

—(g, g, &=6,,(gigi4g() =G~Gk( G(—kGJ(+6((61k,

(Al 1)

APPENDIX B: DISPLACEMENT OPERATORSAND FOURIER TRANSFQRMATIONS

In this appendix we consider, for generality, a set of nHermitian operators g ', i = 1, . . . , n, satisfying theClifford algebra anticommutation rules

and so on, which is the expression of Wick's theorem. Inthe expectation value of 2n fields there are (2n —1)!!terms and those corresponding to odd permutations comein with a negative sign, as one would expect for a fermiontheory. Similarly, closed loops in the Feynman diagramsalso come in with a minus sign.

As a consequence [&' 0™]=&( (81)

(A6)

Since g =0, one only needs to define the integral of g.For real variables, we define

fgdg=i, (A7)

since it is invariant under complex conjugation.The rule for a change of variables is similar to the usu-

al one, the only difference being that one must divide bythe Jacobian of the transformation instead of multiplyingby it, i.e.,

In the case of spin —,', considered in the text, one hasn =3. In this Appendix and in the next, however, we willconsider the case of general n."

The generator of the antisymmetrized products ofthose operators, needed for the antisymmetric or Weylrepresentation of an operator in phase space, is given bythe operator

D(p) =e~'~, (82)I 1where p' are real Grassmann numbers and g.p=g(g'p'.

This operator is unitary since one has

ff(0)d "0=ff(PS )) (A8) (p) =D '(p) =D( —p) . (83)

since Eq. (A7) holds for both the old and new variables.Perturbation theory is done choosing a quadratic Ham-

iltonian for the free theory and it relies, using the func-tional integration approach, on Gaussian integrals andtheir properties. Making a shift in the integration vari-ables, one completes the square, and by a change of vari-ables one determines the prefactor. The basic Gaussianintegral is then given by

f exp(2$ 6 'g+g p)dg=P( —6 ')exp(](M 6(](,),(A9)

where G is an even-dimensional antisymmetric matrix,and dg=dg„. . dg]. The Pfaff]an P(A) of a matrix Ais defined by

1 1

(~/2)] g ( 1 ) ~o(1br(2 ]

It has the important property

'(p)gL) (p) =g+pand is the analog of the displacement operator

eaa —a a

(84)

(85)

etc. One then has

a a —a*a =—[(a]a2+a2a] ) —(a2a]+a]ai)]

used in quantum mechanics, field theory, or quantum op-tics. For bosons a, a* are complex numbers and for fer-mions they are complex Grassmann variables. This con-nection is simply established, starting from Eq. (85) andusing a decomposition of real and imaginary parts of theform

1 . Ia = —(a, +ia2), a= —(a]+iaz),v'2 2

X A ~(3)(y(4) 0.(N 1)0[Ã)

(A 10)+ —,'[(a]a]—a]a] )+(a~a~ —a2a2)] . (86)


For bosons, it is the first term which survives. CallingX 2 =P d ~i =X, a2=P, one finds

a a —a"a =i(XP PX—)

giving the displacement operator in terms of the di-mensionless operators X,P. For an harmonic oscillatorwith mass m and natural frequency ~o, they areX=(meso/R)' x and P=P/(mcoofi)' in terms of theusual coordinate and momentum.

For fermions, it is the second term which survives andone has

a, n, .

g(C)= f fg(C')l l+(0' 0—) p]d(P')*dp

= f fg(g')e'~ ~'t'd(g')*dp

= fg(p)e ~t'dp,

with g(p) given by Eq. (812).

(814)

This can be easily checked noting that, for Grassmannvariables, the delta function is simply 5(g' —g)

i—(g' —g), since one has

i —fg(g')(g' —g)dg'=g(g) . (813)

In the case of several variables, this can be rewritten as

There is a term like this for each degree of freedom, lead-ing to a theory with an even number of operators, associ-ated in pairs, since for each creation operator there is thecorresponding destruction operator. In Eq. (88), the ex-tension to an odd number of operators offers no di%culty,contrarily to what happens in the case of Eq. (87) whereit cannot be done. In fact, the extension of anticommuta-tion relations to an odd number of operators offers nodifhculty, contrarily to what happens in the case of com-mutators.

The multiplication law for the displacement operators1S

' 'D(p2+pi)

showing that the operators D(p) are closed under multi-plication. Since D(p)=e~ t' is the g'enerator of the an-

Itisymmetrized products of the operators g', the symbol ofsome operator g(g), expressed as

g(g) = fg(p)e ~t'dp, (810)

where dp=dp„. dp&, i.e., the function representing itin phase space, in the antisymmetric or Weyl representa-tion, is simply obtained replacing the operator g by theclassical variable g, and one has

g(g)= fg(p)e &'~dp . (811)

This is a Fourier transformation formula, g(p) beingthe Fourier transform of g(g). The difference betweenEqs. (810) and (811) is that in (810) the exponential is anoperator, whereas in Eq. (811) it is not. Fourier invertingEq. (811)one finds

g(p)= fg(g)e~t'd*g, (812)

where

1 )n(n —i)/2dg

APPENDIX C: PHASE-SPACE REPRESENTATIONOF QUANTUM MECHANICS

As discussed by Berezin and Marinov, once one hasestablished the connection between quantum mechanicaloperators and functions defined in phase space, one mustknow how to obtain the function representing the prod-uct of two or more operators and the trace of an operatorin terms of the functions representing them. This isnecessary for the construction of the functional integralfor the evolution operator, writing it as the product ofvery many infinitesimal evolution operators and for thepartition function which envolves the trace in itsdefinition.

1. Product of operators

Using the multiplication law of Eq. (89) one verifiesthat the functions representing the product of two opera-tors is given by

g(g)= f f exp[ —g (p2+p, )

——,'p2 p, ]g2(p2)dp2g, (p, )dp,

in terms of the Fourier transforms of the functionsrepresenting those operators in phase space. Using Eq.(812) to write this expression in terms of the functionsthemselves, and doing the gaussian integration on p, and

pz using Eq. (A9), one arrives at

—2 —1 0

FIG. 4. Iteration scheme for the multiplication of % opera-tors (X=6). FICx. 5. Graph of &f (t }.

FINITE- TEMPERATURE REAL-TIME FIELD THEORIES FOR. . . 7X89

I )n (n + ()/2g(g) = f fg2($2)g, (g, )exp[ —2(g) $2+$2 g+g g, )]dg, dg2 . (C2)

Iterating this expression as sketched in Fig. 4, one finds that the function representing the product g =gN . g1, of Noperators is given by

I )n(n + ()/2g(g)= gN N gN —1 N —1 g1 1

N

Xexp —2 g (g, v], , +g, +,.g;+g; g;) g dq„dgk, (C3)

where we have defined g)v+) =g. Since the functions g;(g;) do not depend on the variables q, the integrals over themcan be immediately performed. However, in order to use Eq. (A9) for the calculation of this Gaussian integral, oneshould have an even number of operators. Then discuss the case of an even number N of operators first.

The exponential factor in Eq. (C3) can be rewritten as

exp —2 g (g, g, +, +g;+) g;+q; g;) =exp(q Ari+2q Bg 2g~—g), . (C4)

where

00

0

0

0000

000

0

A=00 0

shown for N =8, with

A is an N XN matrix of the type

CT 0 0 00

(C5)

with

0e

1

In order to apply Eq. (A9), it is necessary to evaluateA ' and B A 'B. They are matrices of the type

0 1 0 0+=0 0 ' 1 0

(C7)

and B is an N X (X+ I ) matrix of the type

8 A '8=0 +0

—1+o.++o.1+0 +0

+T+e T

1 —o.+ —o.

0 +0—1+0-++�-

o+�T T

1 0 01 0 0—o-++o-

+T

e+ —e

e+ —e

0

(C8)

respectively, with

0

Using the identity

A 11 A 12det

21 22=(detA22)det( 3 )( —A, 2 3 22'A2) ) (C9)

one finds that detA =1.The product of an even number N of operators is finally given by


~ n(n +1)/2

f '' fgN(4)'2n /2

Xexp(g) . gN) d g) d gN (C10).

0N ~

2. Trace of operators

In general, the trace of the product of an odd numberof Clifford operators g

' is zero, i.e., one has

Trg' . . (~=0 (Cl 1)

for p odd, except for the product of all of them, whentheir number n is odd. As known, Clifford algebras withan odd or an even number of generators must be treateddifferently.

The trace of an even number of g' is given by

Tr1=d,

The prefactor can be most easily checked choosing

1gk(gk ) — 2

(n —))/2 kk 0kl

which, as we will see in the following, is the odd represen-tation for the identity.

The case of the product of an odd number of operatorscan be treated in two different manners. One consists inexcluding one of the operators, the last one, for example,using Eq. (C10) for the product of all the others (sincetheir number is even) and using then Eq. (Cl) for theremaining product. Alternatively, one can introduce theidentity as an extra operator, before the erst or after thelast, for example, and then use Eq. (C10) for the productof all of them.

are not allowed, we conclude that the terms contributingto the trace of an operator are the identity and the prod-uct of all the g', when n is odd. In this case the productof all the generators must be a multiple of the identity,since it commutes with all of them. Its square is

1 )n (n —) )/2(pl. . . gn)2

2n

One must then have~ n (n —1)/2g). . . gn

2n /2 (C13)

where P are the permutations of 1, . . . , n and o. its signa-ture, one has

2n/2

n(n —) )/2 ~)

and

g( )—+2n/2)n(n+))/2

where the two signs + refer to the two possible left orright orientations of those generators.

For the identity represented by the operator 1, one hasg(g)=1. Using Eq. (812) one findsg(p)=( i)"p) .—p".Similarly, for the identity represented by

.2n /2

Trg 'g =—5) (C12)One concludes then that the trace of a general operator gis given by

TCY' 0"0'=, (&) &„—&(,& q+&(q&, »22 Trg(g)=d g(0)+, „g(0)1(C14)

and so on, where d is the dimension of the irreduciblerepresentation given by d =2, where m is the integergiven by n =2m + 1 or n =2m for n odd or even, respec-tively. In the case of spins —,', we have n = 3 and d =2, aswe should. Equation (C12) shows that to have a nonzeroresu1t the Clifford operators must occur in pairs. Thosepairs can be brought together commuting one of theoperators with the others in between, introducing minussigns. The square of a single Clifford operator is —,', andone eventually arrives at a multiple of the identity. Thedelta functions in Eq. (C12) keep track of the number ofpermutations. Since when making the transcription fromoperators to functions, terms having operators repeated

~ n (n —1)/2=d g(0)+ „/2 „ fg(g)dg (C15)

r

n/2 ~ n(n +))/2 ga 1 1

E

(C16)

1 ( —1)"2n/2

&

n (n —1)/2 (C17)

Since the product g, g„ is a multiple of the identity,

From these equations and Eqs. (89) and (810), one ob-tains that


to each operator corresponds two different functionsrepresenting it: one with an even and the other with anodd number of operators. The existence of two terms inEqs. (C14) and (C15) is a consequence of this. For con-venience, we will use the even representation for g. Con-sequently, g is odd. From Eq. (C16) it follows that

g(a) =Tr [D(2a)+D( —2a)]g,2d

(C18)

expressing g (a) in terms of g.The trace of the product of several operators is ob-

tained combining Eq. (C10) with Eqs. (C14) or (C15).When using the even representation, Eqs. (C14) and (C15)for the trace reduce to

1 2 3 4 5 6Trg(g) =dg(0), (C19)

and the trace of an even number of operators is simplygiven by Eq. (C10) taking /=0, and multiplying by d.

APPENDIX D: THE CONTINUUM LIMIT

In this appendix we take the continuum limit of thediscrete approximation for the functional integral forspin —,, i.e., we discuss the transition from Eq. (3.9) toEqs. (3.10) and (3.11) in the main text. Let us considerfirst the g dependent term of the exponent in Eq. (3.9).

FIG. 6. Layout of terms, for the continuum limit {X=6).

This term is

2k. ( —01+02+ ' —4 -1+4 )

giving, in the continuum limit

g [ g(r, )—+g(tf)],since

201+02+ kN —1+ 20N T[( 01+02) ( 42+03)+ +(gN —2 CN —1) (gN —1 0N)]1 tk tJ

=-,' g ( —1)" 'J dr f p(r) dr~0,k=1

where the function P(t) defined by

( 1)k —I

p(r) =2

for tk & t & tk+ „is a rapidly varying function, as shown in Fig. 5, leading to a vanishing result in the continuum limit.This is similar to the use of the Euler-MacLaurin summation formula for which one would take

p(t) = ——+2 tk+, —tk

The other term in the exponent of Eq. (3.9), depending on the fields g, can be handled in a suitable two-dimensionalgeneralization of the method just used. Considering the terms with i &j, which is half of the total result, and writing itin the form

N —1 N N —2 N —1( 1)J

—i

( ) CJ'0 = r X 4 (kJ'0 CJ+1'k PJ'0 +1+PJ+1'r +1)j=1 i =j+] J=1 E =J+1

N —1——'g + g ( —1)'g +—'g

X —1 N —1

+l i&1+ X ( ——1)' "0J+-,'0N 0N—

—.' X 0J (CJ+1 —0J)——.'01.0N

J =2(D2)

where use of (g ) =0 was made. The first term in the right-hand side vanishes in the continuum limit since it is of theform

,)

82$(r) g(t')d d,

where the function p(t, t')=( —1) ' for t, &r &t, +, , and t &t'&t, +, is a rapidly varying function. The second andthird terms are of the type considered before, vanishing also in the continuum limit. The last two terms give then

I

7192 VITOR ROCHA VIEIRA

—,' —f g dt ,'—g(—t,) g(t&)tg

l

in the continuum limit.Analysis of the different terms involved and their weight, can be done using Fig. 6. As a result, the action for the

functional integral for spin —, is given by

is= —,' f—g dt i—f &(t)dt+ ,'g(tI)—g(t,)+g(t, ) g+g. g(t/) . (D3)

This derivation corrects a mistake in Berezin s and Marinov s paper where the factor —, in g(t&).g(t, ) is missing. This 1s

important for the correct determination of the boundary conditions for the equations of motion, discussed and used ex-tensively in the main text. Defining

lim dg, dgtv1

(D4)+—+ co

the functional integral for spin —, is given by

K(g, t&, t, )= fX)ge's

with the action given by Eq. (D3).

(D5)

APPENDIX K: DERIVATION OF THE GENERATING FUNCTIONAI. S

In this appendix, we derive the free generating functionals for the different types of formalisms discussed in the maintext. Using Eqs. (C10) and (C19) for the product of four operators and for the trace, Eqs. (4.16) and (4.23) for the func-tions representing the evolution operators and the density matrices, one concludes that the free generating functional ofEq. (5.1), in the main text, corresponding to the use of the path C in the complex time plane, shown in Fig. 2, is givenby

0

1 11

Z[v, t', v, t ]= „„,, f f f

fdic,

d(2d(3d04exp 0102030401 0 —1

1 —1 1 0 04

Xcosh (p —tr) —exp A+ g (2ig~v l ~F40&) cos exp —A'+ g [2p~(p') +iU A+3]a a

T

Xcosh exp A'+ g [2ipq(v') l ~+&02] cos2 2

exp A + g (—2(,P —i U PP, ) (El)

1 cosh[(/3 —o. )(H/2)]cosh(oH/2) 2 HTA + A, + A

26 cosh(tt3H/2) 2

f f f fdcid&2d(3 04exp (01020304

—iU 0'.

—V' —1a

iUlV1

+2( $1/2/3(4)3 p

Lv

(E2)

where the notation follows from Eqs. (4.17)—(4.19) and (4.24) —(4.26).This is a Gaussian integral. Either using Eq. (A9) or simply minimizing the exponent, one finds that the Gaussian in-

tegral is given by

Z[ ,vt'ii, vp]=e pxA+A'+A —A'+( Piv'P'iv)Q—EV

(E3)


where Q is the matrix defined by

1

(1+U )(1+V„V' )

—(V +V' )+iU (1+V V' )

—(1 iU—)(1+V )

—(1—V' )(1+V )

—(1—V' )(1+iU )

—(1—V )(1+iU )

—V (1+U )

—(1+V' )(1—V )

(1 iU—)(1—V )

(1+V' )(1 iU—)

(1 i—U )

—(1+V )(1+iU ) —(V + V' )—iU (1+V V' ) (1—V' )(1 iU—)

—(1+iU )—(1+V' )(1+iU )

—V' (1+U )

(E4)

is the inverse of the matrix in Eq. (E2), as can be verified by inspection. The prefactor is simply one, as expected fromthe normalization of the generating functional.

This is the expression corresponding to the general path of Fig. 2, with source fields in all the branches. To get theTFD and CTPGF formalisms one takes v= v'=0. The generating functional collapses to

nHTZ[@.', p, ]=exp A —A'+ g (p —(p, ') )cos

a

PaH . aHT—tanh2

+2

+& tane (P/2 —o. )aH

PaHcosh

e (P/2 —a )aH

PaHcosh

PaH—tanh2

aHT—i tan2

Substituting the expressions for A, A' and p, (p') one finally gets

(E5)

Z[p', p]=exp f dt f dt'[p, (t)(p') (t)]G (t t')—I

where

G (t t')—(E6)

1 1e(t —t )— e(t t')—e

—PaH+ 1 1+eP He (P/2 —o )aH

aH2 cosh

e (P/2 —a )aH

2-.hP H2

1 e(t t') ——1 e(t' —t)PaH paH+ 1

e—iaH(t —t')

(E7)

The Green's function G (t t') and 6 (t —t') satisfy—the same diff'erential equation with dift'erent boundary condi-tions. The diagonal elements of the matrix in Eq. (E5) replace the trigonometric functions appearing in 6 (t t ) by-the hyperbolic functions of 6 (t —t').

In the main text we discuss the need of considering the imaginary branch in the calculation of equilibrium correlationfunctions. The CTPGF formalism is generalized, keeping the source fields in the imaginary branch of Fig. 1, for o =O.Besides the terms already contained in Eq. (E7), giving the real-time components of the three component formalism andthe term A, leading to Eq. (4.29), giving the imaginary time component, one finds the additional exponential factor


exp ' gi cosaHT '

r

[—a( —)a] & e

—f(aHT/2) & a( 1 1 ) e((aHT/2)(p')

=exp —i dt e '[(tt (t)((M') (t)]

1 f d~ e0

1v (r)

1+e

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Phys. —JETP 20, 1018 (1965)].4R. A. Craig, J. Math. Phys. 9, 605 (1968).5R. Mills, Propagators for Many Particle-Systems (Gordon and

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~4C. de Dominicis, in Dynamical Critical Phenomena and Relat-ed Topics, Vol. 104 of Lecture Votes in Physics, edited by C. P.Enz (Springer-Verlag, Berlin, 1979).P. C. Martin, E. D; Siggia, and H. A. Rose, Phys. Rev. A 8,423 (1973); U. Deker and F. Haake, ibid. 11, 2043 (1975); C.P. Enz and L. Garrido, ibid. 14, 1258 (1976).M. Suzuki, J. Stat. Phys. 42, 1047 (1986).

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~The temperature T should not be confused with the timedifference T = tf —t; used extensively in this paper, since thetemperature dependence will be expressed in terms ofP= 1/kt( T, instead.

35It is preferable, then, to use &—pA' in the evolution operatorsinstead of & only. This affects the form of the (anti) periodicboundary conditions in the imaginary time direction, accord-ing to factors of e —~". See Refs. 18 and 19.H. Matsumoto, Y. Nakano, H. Umezawa, F. Mancini, and M.Marinaro, Prog. Theor. Phys. 70, 599 (1983).

37F. A. Berezin, The Method of Second Quantization (Academic,New York, 1966).

38F. A. Berezin and M. S. Marinov, Ann. Phys. 104, 336 (1977).They should be compared to the expression

tHO(tf- t )

K (cxf tf cubit' ) ( ctf lie

—itoO(t —I. )=exp afe ' a;

f —iroo(t —t. )

+afi dt e ' g(t)t

+i f dt's (t)e 'a;t

—f dt f dt'rt*(t)e ' q(t')l t

for the holornorphic representation of the evolution operator,i.e., for its matrix elements in the basis of the non-normalizedcoherent states

)(a) =e" ' 'D(a))0) =e' ~0)

in the case of free bosons or ferrnions with a Hamiltoniangiven by

&0{t) =boa a —q*{t)a —a q{t)

as shown for example in Ref. 10.Y. Fujimoto, R. Grigjanis, and R. Kobes, Prog. Theor. Phys.73, 434 (1985); Y. Fujimoto and R. Grigjanis, Z. Phys. C 28,395 (1985).H. Matsumoto, I. Ojima, and H. Umezawa, Ann. Phys. (N.Y.)152, 348 (1984); H. Matsumoto, Y. Nakano, and H.Umezawa, Phys. Rev. D 29, 1116(1984).I. R. Pimentel and V. R. Vieira (unpublished).T. Arimitsu and H. Umezawa, Prog. Theor. Phys. 74, 429(1985).

440ur convention differs from the one used by Berezin and


Marinov, Ref. 38, where this integral is normalized to 1.4~In their paper, Berezin and Marinov, Ref. 38, besides the gen-

eral case, consider the particular case n =3, for the evolutionoperator of a nonrelativistic spin in a magnetic field, withoutsources, and the case n =5, in the context of the relativistic

Dirac equation.46See, for example, Handbook of Mathematical Functions, egitetl

by M. Abramowitz and I. A. Stegun (Dover, New York,1972}.

Finite-temperature real-time field theories for spin 1/2

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