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Soviet Physics Uspekhi

Analytical methods of calculating correlation functions in quantum statistical physics

Михаил Федорович Сарры

Published Nov 30, 1991• 14 citations
Publisher

Abstract

A brief but clear and complete account is given for two analytical methods of calculating correlation functions in quantum statistical physics from first principles—the widely used method of two-time temperature Green's functions (the GF method) and a new, "direct algebraic" (DA) method. The mathematical and technical clarity and simplicity of the DA method and its resulting practical value are demonstrated for the five most widely used models in quantum statistical physics. Since the DA method is an exactly self-consistent method (in the sense that the expansion coefficients in the equations of motion are chosen from the requirement that the Jacobi operator identity be satisfied exactly), it in principle affords the possibility of an internal check, which is not possible in the GF method. Like the GF method, the DA method permits calculation of the spectra of possible elementary excitations and, hence, of the density of single-particle energy states corresponding to them.

DOI
10.1070/pu1991v034n11abeh002482

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•Feb 1, 1963•0 citations
A brief but clear and complete account is given for two analytical methods of calculating correlation functions in quantum statistical physics from first principles—the widely used method of two-time temperature Green's functions (the GF method) and a new, "direct algebraic" (DA) method. The mathematical and technical clarity and simplicity of the DA method and its resulting practical value are demonstrated for the five most widely used models in quantum statistical physics. Since the DA method is an exactly self-consistent method (in the sense that the expansion coefficients in the equations of motion are chosen from the requirement that the Jacobi operator identity be satisfied exactly), it in principle affords the possibility of an internal check, which is not possible in the GF method. Like the GF method, the DA method permits calculation of the spectra of possible elementary excitations and, hence, of the density of single-particle energy states corresponding to them.
We extend the formalism of the thermodynamic two-time Green's functions to nonextensive quantum statistical mechanics. Working in the optimal Lagrangian multiplier representation, the q -spectral properties and the methods for a direct calculation of the two-time q Green's functions and the related q -spectral density ( q measures the nonextensivity degree) for two generic operators are presented in strict analogy with the extensive (q=1) counterpart. Some emphasis is devoted to the nonextensive version of the less known spectral density method whose effectiveness in exploring equilibrium and transport properties of a wide variety of systems has been well established in conventional classical and quantum many-body physics. To check how both the equations of motion and the spectral density methods work to study the q -induced nonextensivity effects in nontrivial many-body problems, we focus on the equilibrium properties of a second-quantized model for a high-density Bose gas with strong attraction between particles for which exact results exist in extensive conditions. Remarkably, the contributions to several thermodynamic quantities of the q -induced nonextensivity close to the extensive regime are explicitly calculated in the low-temperature regime by overcoming the calculation of the q grand-partition 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.
The quantum dynamics of correlated fermionic or bosonic many-body systems following external excitation can be successfully studied using nonequilibrium Green functions (NEGFs) or reduced density matrix methods. Approximations are introduced via a proper choice of the many-particle self-energy or decoupling of the BBGKY-hierarchy, respectively. These approximations are based on Feynman's diagram approaches or on cluster expansions into single-particle and correlation operators. In a recent paper [E. Schroedter, J.-P. Joost, and M. Bonitz, Condens. Matter Phys. 25, 23401 (2022)] we presented a different approach in which, instead of equations of motion for the many-particle NEGF (or density operators), equations for the correlation functions of fluctuations are analyzed. In particular, we derived the stochastic GWGWGW and polarization approximations that are closely related to the nonequilibrium GWGWGW approximation. Here, we extend this approach to the computation of two-time observables depending on the specific ordering of the underlying operators. In particular, we apply this extension to the calculation of the density correlation function and dynamic structure factor of correlated Hubbard clusters in and out of equilibrium.
The quantum inverse scattering method is a means of finding exact solutions of two-dimensional models in quantum field theory and statistical physics (such as the sine-Gordon equation or the quantum non-linear Schrödinger equation). These models are the subject of much attention amongst physicists and mathematicians. The present work is an introduction to this important and exciting area. It consists of four parts. The first deals with the Bethe ansatz and calculation of physical quantities. The authors then tackle the theory of the quantum inverse scattering method before applying it in the second half of the book to the calculation of correlation functions. This is one of the most important applications of the method and the authors have made significant contributions to the area. Here they describe some of the most recent and general approaches and include some new results. The book will be essential reading for all mathematical physicists working in field theory and statistical physics.
Based on the Green's function (GF) equation-of-motion formalism, we develop a method to expand the double-time Green's function into Taylor series of the parameter $\ensuremath{\lambda}$ in the Hamiltonian $H={H}_{0}+\ensuremath{\lambda}{H}_{1}$. Here H0{H}_{0}H0​ is the exactly solvable part and H1{H}_{1}H1​ is regarded as the perturbation. To restore the analytical structure of GF, we use the continued fraction to do resummation for the obtained series. The problem of zero-temperature divergence is identified and remedied by the self-consistent series expansion. To demonstrate the implementation of this method, we carry out the weak- as well as the strong-coupling expansion for the Anderson impurity model to order ${\ensuremath{\lambda}}^{2}$. Improved result for the local density of states is obtained by self-consistent second-order strong-coupling expansion and continued fraction resummation.