Kinetic coecients in the formalism of time-dependent Greens functions at nite temperature Viacheslav Krivorol1Michail Nalimov12

2025-05-04 0 0 473.81KB 18 页 10玖币

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Kinetic coeﬃcients in the formalism of time-dependent

Green’s functions at ﬁnite temperature

Viacheslav Krivorol1,∗Michail Nalimov1,2,†

1Saint-Petersburg State University, 7/9 Universitetskaya Emb.,

St Petersburg 199034, Russia

2Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear

Research, 6 Joliot-Curie, Dubna, Moscow region, 141980, Russia

Abstract

We discuss the microscopical justiﬁcation of dissipation in model nonrelativistic Fermi

and Bose systems with weak local interactions above phase transitions. The dynamics

of equilibrium ﬂuctuations are considered in Keldysh – Schwinger framework. We show

that the dissipation is related to pinch singularities of the diagram technique. Using

Dyson – Schwinger equation and the two-loop approximation we deﬁne and calculate the

attenuation parameter which is related to exponentiality of Green’s functions decay. We

show that the attenuation parameter is the microscopic analogue of the Onsager kinetic

coeﬃcient and it is related to attenuation in the excitation spectrum.

1 Introduction

Theoretical description of dissipation phenomena and dynamics of ﬂuctuations in quantum

many-body systems is an interesting and important problem of modern theoretical and experi-

mental physics [1–4]. Physically, the nature of dissipation is related to interactions between the

system and its environment. However, to fully understand this phenomenon, a more detailed

theoretical consideration is required. Historically, the ﬁrst attempts to describe dissipation from

the ﬁrst principles of quantum mechanics apparently were made by Feynman and Vernon [5,6].

The proposed approach was to couple the system (ﬁnite-dimensional in the simplest case) with

a model environment, for example, with a system of harmonic oscillators [7]. In some cases,

dissipation eﬀects in systems with a model environment can be studied by obtaining the exact

solution. Further research in this area led to the creation of open quantum systems theory [8].

The main tool of this formalism is the so-called Lindblad equation [9–11]. It is a generalization

of the standard Neumann equation for the density matrix to the case of Markovian dissipative

and non-Hamiltonian evolution [12]. This equation is the von Neumann equation with addi-

tional terms. These terms model the interactions between the system and the environment.

The standard problems of this formalism are the choice of the explicit form of additional terms

for various models and the complexity of the microscopic justiﬁcation of this choice.

As is known, dissipation is standardly described in terms of kinetic coeﬃcients. The stan-

dard result on the kinetic coeﬃcients microscopic structure are the Green – Kubo formu-

las [1,2,13,14]. The correlators arising within this approach have an extremely complex

∗E-mail: v.a.krivorol@gmail.com

†E-mail: m.nalimov@spbu.ru

arXiv:2210.14281v1 [cond-mat.stat-mech] 25 Oct 2022

structure and their calculation is usually available only within the framework of numerical

simulation [15–17]. The detailed derivation of the general formula for the kinetic coeﬃcients

of linear hydrodynamics in the Mori’s projecting operators can be found in [1,2]. The general-

ization of Mori’s method to a nonlinear case was done in [18], but this approach has not been

widely applied yet. Note that the methods of the linear response theory (in particular including

formulas of the Green – Kubo type) applied to quantum systems use the method of the tem-

perature Green’s functions analytic continuation [19]. However, in complicated situations using

purely temperature-based methods seems less clear compared to the time-dependent Green’s

functions at ﬁnite temperature [20], which directly take into account the dynamics.

We are interested in constructing hydrodynamics from the ﬁrst principles and microscopic

Hamiltonian using ﬁeld-theoretical methods [21]. The motivation for this study is the standard

problem of critical dynamics associated with the ambiguity of choosing the correct system of

phenomenological hydrodynamic equations (the most well-known family of models includes

A, B, C . . . models) that describe the relaxation of the order parameter [22]. Sometimes the

choice between phenomenological models is quite diﬃcult, and the question arises whether it

is possible to build such a model based on the microscopic picture. For example, in work [23]

for λ-transition in Bose system it was shown from microscopic consideration1that the simplest

model Ais the correct hydrodynamic model, while phenomenological considerations pointed to

a more complex Fmodel. Following the ideas of paper [23], our goal is to generalize this result

to the case of an arbitrary equilibrium state.

In this paper we study the structure of kinetic coeﬃcients in quantum many-body systems

and calculate them. To demonstrate the main ideas, we consider the dynamics of equilibrium

ﬂuctuations in a simpliﬁed Fermi or Bose system with weak local structureless interactions.

A natural tool for constructing a perturbation theory in this case is the formalism of time-

dependent Green’s functions at a ﬁnite temperature. The dynamics of these Green’s functions

is given on the Keldysh – Schwinger contour. The main ideas of this framework and the

relevant literature can be found in [2,20,25–32]. It was found that, starting from the second

order of the perturbation theory, the so-called “pinch” singularities occur (singularities on

large time scales of a special form) [19,33–37] for the time-dependent Green’s functions at a

ﬁnite temperature. These singularities are speciﬁc for quantum ﬁeld theory2. The diagrams

containing the singularities were regularized and calculated in the two-loop approximation. It

was noted that this procedure followed by “dressing” the regularization parameter according

to the Dyson equation makes it possible to prove the existence of exponential in time decay of

the total two-point Green’s function. That was not observed at the level of propagators. That

is, the presence of pinch singularities in this system leads to attenuation in the quasiparticle

spectrum. In this paper, in the two-loop approximation we show that it is possible to explicitly

calculate the exponential decay factor as a function of temperature and chemical potential.

The article is organized as follows. In section 2we discuss the general technique of time-

dependent Green’s functions at a ﬁnite temperature for the microscopic model Hamiltonian of

a Fermi or Bose system with weak local interactions. In section 3we discuss the dissipation

emergence mechanism for this system using the Dyson equation. The attenuation parameter

is determined and calculated in the two-loop approximation. In Appendix A we discuss some

technical details of the asymptotic analysis of integrals arising from the “dressing” procedure.

In Appendix B we give the ﬁrst coeﬃcients of the Taylor series expansion in terms of the

frequencies of some Feynman diagrams.

1The similar analysis for spin systems on a cubic lattice was done in [24].

2Note that the similar singularities of the Keldysh perturbation theory were noticed when describing the

dynamics of relativistic particles on a curved space-time background [38–41].

2 Time-dependent Green’s functions at ﬁnite tempera-

ture

We consider non-relativistic Fermi or Bose many-body system with weak local interactions

“density – density” type. To illustrate of our ideas, the speciﬁc form of the interaction potential

is not important, but the calculation of Feynman diagrams becomes more complicated in a

general situation. Thus, we work in the local interactions approximation. The generalization of

our technique for arbitrary non-local potential is straightforward. We suppose that the system

is homogeneous and isotropic in space. The well-known standard Hamiltonian [42,43] of this

system is given by3

H=Zdxˆ

ψ+(x, t)−∆

2m−µˆ

ψ(x, t) + gZdxˆ

ψ+(x, t)ˆ

ψ(x, t)ˆ

ψ+(x, t)ˆ

ψ(x, t).(1)

Here mis the mass of particles, ˆ

ψand ˆ

ψ+are fermion or boson ﬁeld operators, gis the coupling

constant4,µis the chemical potential. Unless otherwise indicated, all integrals in this paper are

assumed to be taken with inﬁnite limits. The sum over the spin indices of the ﬁeld operators

is implied.

We describe the dynamics of ﬂuctuations in the system within the formalism of time-

dependent Green’s functions at the ﬁnite temperature. In this framework, the standard objects

are 2-npoint Green’s functions

G2n= SpTˆ

ψ(x2n, t2n). . . ˆ

ψ(xn, tn)ˆ

ψ+(xn−1, tn−1). . . ˆ

ψ+(x2, t2)ˆ

ψ+(x1, t1)ˆρ,(2)

where Sp is the trace operation in the quantum mechanical sense, ˆρis the density operator

which describes the distribution in the system at an initial moment of time tin. Quantum ﬁelds

are considered in the Heisenberg representation:

ψ(x, tk) = eiˆ

H(tk−tin)ˆ

ψ(x)e−iˆ

H(tk−tin), k = 1,2. . . 2n, (3)

where ˆ

ψ(x) is an operator equal to the ﬁeld operator in the Schr¨odinger representation at

initial moment of time tin. It is convenient not to ﬁxing the total number of particles, thus we

assume that density matrix ˆρdescribes the big canonical ensemble with an inverse temperature

β= 1/T and the chemical potential µ. We assume that the temperature Tand the chemical

potential µdeﬁne a system without anomalous averages. Further consideration is only valid in

this situation. The diagram technique for Green functions (2) is discussed in works [23,44,45] in

terms of the operator formalism [46]. This framework is complicated, so we take the functional

integral point of view. The diagram technique may be obtained by considering the following

generating functional [47]

ZDψ+Dψ eiS+A+ψ+Aψ+.(4)

Here Aand A+are the ﬁeld sources, the symbol Ddenotes the functional integration, ψand

ψ+are fermionic or bosonic ﬁelds. The action Scan be written in the form

S=Z

dt"ψ+(x, t)i∂t+∆

2m+µψ(x, t)−gψ+(x, t)ψ(x, t)ψ+(x, t)ψ(x, t)#,(5)

where Cis the Keldysh – Schwinger contour [48–50] (see Figure 1). The form of the action

3In this paper ~= 1 and Boltzmann constant kB= 1.

4It could be negative in Fermi systems.

tin −iβ

tin tf

Figure 1: Keldysh – Schwinger contour

(5) reﬂects the fact that there is a set of ﬁelds ψR, ψA, ψT(and the conjugates), indices R

(for retarded), A(for advanced), T(for temperature) indicate the straight-line section of the

contour on which the corresponding ﬁelds are deﬁned. The following boundary conditions are

required at the ends of the contour

ψR(tf) = ψA(tf), ψA(tin) = ψT(tin), ψR(tin) = ±ψT(tin −iβ).(6)

Hereinafter we use the shorthand notation ±and ∓, where sign in the upper position is used

for bosons and sign in the lower position is used for fermions. The ﬁrst two conditions (6) are

the continuity conditions of the ﬁelds on the Keldysh – Schwinger contour. The third condition

is the standard anti-symmetry of fermionic ﬁelds and symmetry of bosonic ﬁelds similar to the

one in the theory of temperature Green functions [46].

By h. . .i0we denote the averaging of ﬁelds with gaussian weight exp (−S0). Here S0is the

part of the action (5), quadratic in ﬁelds, and multiplied by −i. The propagators

hψi(x, t)ψ+

j(x0, t0)i0≡Gij+(x, t, x0, t0)

are deﬁned as the solutions of the matrix equation [46]

∂t−i∆

2m−iµ

hψRψ+

Ri0hψRψ+

Ai0hψRψ+

Ti0

hψAψ+

Ri0hψAψ+

Ai0hψAψ+

Ti0

hψTψ+

Ri0hψTψ+

Ai0hψTψ+

Ti0

=δ(t−t0)δ(x−x0)



100

0−1 0

001

(7)

which is consistent with the boundary conditions (6). The propagators carry the indices

i, j ∈ {R, A, T }

of the corresponding ﬁelds. Solving (7) in time – momentum space, we obtain

GRR+=θ(t−t0)±n(ε)e−iε(t−t0), GRA+=±n(ε)e−iε(t−t0), GRT +=±n(ε)e−iε(t−t0),

(8)

GAA+=−θ(t0−t)∓n(−ε)e−iε(t−t0), GAR+=∓n(−ε)e−iε(t−t0), GAT +=±n(ε)e−iε(t−t0),

(9)

GT T +=θ(t−t0)±n(ε)e−iε(t−t0), GT R+=∓n(−ε)e−iε(t−t0), GT A+=∓n(−ε)e−iε(t−t0).

(10)

where θ(t−t0) is the Heaviside step function, n(ε) = (eβε ∓1)−1is the average occupation

number of the level with energy ε=p2/2m−µ,pis the momentum (with respect to x−x0).

Following Keldysh method and using the freedom of choice of time moments tin and tfit

is convenient to set the limits tin → −∞, tf→+∞. It is customary to facilitate the use of

the Fourier transform. After passing the limit the system becomes homogeneous in time (see

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Kineticcoecientsintheformalismoftime-dependentGreen'sfunctionsatnitetemperatureViacheslavKrivorol1;*MichailNalimov1;2;1Saint-PetersburgStateUniversity,7/9UniversitetskayaEmb.,StPetersburg199034,Russia2BogoliubovLaboratoryofTheoreticalPhysics,JointInstituteforNuclearResearch,6Joliot-Curie,Dubna,Mo...

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Kinetic coecients in the formalism of time-dependent Greens functions at nite temperature Viacheslav Krivorol1Michail Nalimov12

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