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Zhur. Eksptl'. i Teoret. Fiz.

ON APPLICATION OF QUANTUM FIELD THEORY METHODS TO PROBLEMS OF QUANTUM STATISTICS AT FINITE TEMPERATURES

Alexei A. Abrikosov, L. P. Gor’kov, I. E. Dzyaloshinskiǐ

Published Mar 1, 1959• 37 citations
OpenAlex

Abstract

S>A formulation of the thermodynamic perturbation theory is proposed which permits the application of quantum field theory methods to quantum statistics at finite temperatures. The method is an extension of the Matsubara technique and is based on expansion of the Green's functions in Fourier series in an imaginary time variable. The technique thus arrived at differs from the usual diagram technique for T = 0 in that summation over discrete values of the imaginary frequency is substituted for integration over the frequencies. The analytical properties of Fourier components of Green's functions are investigated. It is shown that due to the possibility of analytic continuation for solution of various kinetic and nonstationary problems, it is sufficient to know the corresponding equilibrium Green's functions. (auth)

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arXiv:1005.0393•Progress of Theoretical Physics•Apr 1, 2010•83 citations
S>A formulation of the thermodynamic perturbation theory is proposed which permits the application of quantum field theory methods to quantum statistics at finite temperatures. The method is an extension of the Matsubara technique and is based on expansion of the Green's functions in Fourier series in an imaginary time variable. The technique thus arrived at differs from the usual diagram technique for T = 0 in that summation over discrete values of the imaginary frequency is substituted for integration over the frequencies. The analytical properties of Fourier components of Green's functions are investigated. It is shown that due to the possibility of analytic continuation for solution of various kinetic and nonstationary problems, it is sufficient to know the corresponding equilibrium Green's functions. (auth)
In the quantum field theory with the vacuum being the ground state the Green functions are the vacuum expectation values of the chronological, retarded or advanced products of the field operators (Bjoken & Drell, 1964; Itzykson & Zuber, 1985; Peskin & Schroeder, 1995). They are the generalized functions of the real time variables ti (and also other spatial coordinates). For the application of the Green function technique to the study of the timeindependent phenomena in equilibrium many-body systems at a finite temperature, the Matsubara imaginary time Green functions were introduced and widely used (Abrikosov et al., 1975; Bruuns & Flensberg, 2004; Haken, 1976). They are the mean values over a statistical ensemble at a finite temperature of the chronological products of the imaginary timedependent operators. Both these types of Green functions are inadequate for the application to the study of the time-dependent phenomena in the many-body systems with a finite density and at a finite temperature, in particular the non-equilibrium systems. For the application to the study of the time-dependent dynamical processes in non-equilibrium many-body systems Keldysh (Keldysh, 1965) has introduced a more general class of timedependent Green functions at finite temperature and density. They are the mean values of the time-ordered products of quantum operators in the Heisenberg picture over statistical ensembles of many-body systems with finite densities and at finite temperatures (which may be non-vanishing). The simplest example is the two-point Green function
In Chapter 9 we applied the second quantization to the Hartree–Fock method. In this chapter we provide the basic aspects of the thermal Green function method, which is convenient for systematic perturbative expansion beyond the mean-field approximation. Several problems in quantum statistical mechanics can be elegantly and compactly described in terms of the so-called Matsubara Green functions (Matsubara (1955); Abrikosov et al. (1963)). Hence, if on one hand, the density matrix description, often used in the previous chapters, has a more direct physical interpretation, on the other hand the Green function method is far more powerful and takes advantage of a handy diagrammatic perturbation expansion. In the following chapters, we rely on the Green function for a more complete treatment of some many-body problems. To be self contained we introduce here this technical chapter with a short presentation of the by now standard Green function method. For a comprehensive treatment we refer the reader to the many books available and in particular to Abrikosov et al. (1963). The Matsubara Green function The main aim of quantum statistical mechanics is the evaluation of the partition function (see Eqs. (6.31) and (6.32)), e.g. in the grand canonical ensemble In the case of quantum gases one can evaluate Ƶ exactly as we did in Chapter 7. In general, the evaluation of the partition function is a difficult task and it is useful to use perturbation theory to obtain approximate solutions. To set up a perturbation theory, one begins by splitting the Hamiltonian as H = H 0 + H i, where H 0 represents the “easy” part, which can be solved exactly. In the second step, one could naturally think of expanding the exponential of the interacting part H i in order to write the partition function as a perturbative series. However, such a procedure requires extreme care due to the non-commutative nature of operators in quantum mechanics. The way of solving this problem is via the “ time ” ordering of operators in each term of the perturbative expansion. It can be shown, as we shall see in the present chapter, that the quantum statistical average of any time-ordered sequence of operators with respect to the unperturbed Hamiltonian H 0 can be expressed in terms of elementary quantum statistical averages of two field operators. Such elementary averages are called Green functions and in this section we introduce the basic definitions concerning them.
Matsubara Green's functions for interacting bosons are expressed as classical statistical averages corresponding to a linear imaginary-time stochastic differential equation. This makes direct numerical simulations applicable to the study of equilibrium quantum properties of bosons in the non-perturbative regime. To verify our results we discuss an oscillator with quartic anharmonicity as a prototype model for an interacting Bose gas. An analytic expression for the characteristic function in a thermal state is derived and a Higgs-type phase transition discussed, which occurs when the oscillator frequency becomes negative.
Matsubara Green's functions for interacting bosons are expressed as classical statistical averages corresponding to a linear imaginary-time stochastic differential equation. This makes direct numerical simulations applicable to the study of equilibrium quantum properties of bosons in the non-perturbative regime. To verify our results we discuss an oscillator with quartic anharmonicity as a prototype model for an interacting Bose gas. An analytic expression for the characteristic function in a thermal state is derived and a Higgs-type phase transition discussed, which occurs when the oscillator frequency becomes negative.
In this article, we present a concise and self-contained introduction to nonequilibrium statistical mechanics with quantum field theory by considering an ensemble of interacting identical bosons or fermions as an example. Readers are assumed to be familiar with the Matsubara formalism of equilibrium statistical mechanics such as Feynman diagrams, the proper self-energy, and Dyson's equation. The aims are threefold: (i) to explain the fundamentals of nonequilibrium quantum field theory as simple as possible on the basis of the knowledge of the equilibrium counterpart; (ii) to elucidate the hierarchy in describing nonequilibrium systems from Dyson's equation on the Keldysh contour to the Navier-Stokes equation in fluid mechanics via quantum transport equations and the Boltzmann equation; (iii) to derive an expression of nonequilibrium entropy that evolves with time. In stage (i), we introduce nonequilibrium Green's function and the self-energy uniquely on the round-trip Keldysh contour, thereby avoiding possible confusions that may arise from defining multiple Green's functions at the very beginning. We try to present the Feynman rules for the perturbation expansion as simple as possible. In particular, we focus on the self-consistent perturbation expansion with the Luttinger-Ward thermodynamic functional, i.e., Baym's Phi-derivable approximation, which has a crucial property for nonequilibrium systems of obeying various conservation laws automatically. We also show how the two-particle correlations can be calculated within the Phi-derivable approximation, i.e., an issue of how to handle the "Bogoliubov-Born-Green-Kirkwood-Yvons (BBGKY) hierarchy".