In this work we will establish the definitions of the Green’s functions in statistical mechanics and their basic properties. These functions depend on double-time and temperature. It can be observed by mean of theier definitions, where we can find the medium values of the operators product, this average is done over the grand canonical ensemble, for instance, (FAB(t − t0) ≡ hAˆ(t)Bˆ(t0)i ou FBA(t − t0) ≡ hBˆ(t0)Aˆ(t)i), with β = 1/kBT, and T is the absolute temperature. The operators involved in theses functions satisfy the Heisenberg equation of motion, which permit us describe the evolution equations for Green’s functions. By mean of spectral representation of the time correlation functions(FAB(t−t0) ≡ hAˆ(t)Bˆ(t0 )i or FBA(t−t0) ≡ hBˆ(t0)Aˆ(t)i), that is done by the introduction of a Fourier transform to change the system from time-space to frequency-space, we can obtain the spectral representations for the retarded Gr, advanced Gr e causal Gc green’s functions. Lastly, we will make the use of retarded green’s function to describe the electrical conductivity of a system of electrons under a external time-dependent electric field, in others words, we will write the electrical conductivity tensor in terms of Gr and, at last, we will calculate the electrical conductivity of a system of electrons and phonons.
In this work we will establish the definitions of the Green’s functions in statistical mechanics and their basic properties. These functions depend on double-time and temperature. It can be observed by mean of theier definitions, where we can find the medium values of the operators product, this average is done over the grand canonical ensemble, for instance, (FAB(t − t0) ≡ hAˆ(t)Bˆ(t0)i ou FBA(t − t0) ≡ hBˆ(t0)Aˆ(t)i), with β = 1/kBT, and T is the absolute temperature. The operators involved in theses functions satisfy the Heisenberg equation of motion, which permit us describe the evolution equations for Green’s functions. By mean of spectral representation of the time correlation functions(FAB(t−t0) ≡ hAˆ(t)Bˆ(t0 )i or FBA(t−t0) ≡ hBˆ(t0)Aˆ(t)i), that is done by the introduction of a Fourier transform to change the system from time-space to frequency-space, we can obtain the spectral representations for the retarded Gr, advanced Gr e causal Gc green’s functions. Lastly, we will make the use of retarded green’s function to describe the electrical conductivity of a system of electrons under a external time-dependent electric field, in others words, we will write the electrical conductivity tensor in terms of Gr and, at last, we will calculate the electrical conductivity of a system of electrons and phonons.
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
Abstract Multiparticle thermodynamic Green’s functions, defined in terms of grand canonical ensemble averages of time-ordered products of creation and annihilation operators, are interpreted as tracing the amplitude for time-developing correlated interacting particle motions taking place in the background of a thermal ensemble. Under equilibrium conditions, time-translational invariance permits the one-particle thermal Green’s function to be represented in terms of a single frequency, leading to a Lehmann spectral representation whose frequency poles describe the energy spectrum. This Green’s function has finite values for both t>t′ and t<t′ (unlike retarded Green’s functions), and the two parts G1> and G1< (respectively) obey a simple proportionality relation that facilitates the introduction of a spectral weight function: It is also interpreted in terms of a periodicity/antiperiodicity property of a modified Green’s function in imaginary time capable of a Fourier series representation with imaginary (Matsubara) frequencies. The analytic continuation from imaginary time to real time is discussed, as are related commutator/anticommutator functions, also retarded/advanced Green’s functions, and the spectral weight sum rule is derived. Statistical thermodynamic information is shown to be embedded in physical features of the one- and two-particle thermodynamic Green’s functions.
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.
Abstract Green-function techniques are among the most versatile tools for investigating quantum many-body systems at finite temperature. Yet, standard textbook treatments often fragment the formalism—separating bosons and fermions, systems with massless and massive particles, or restricting proofs to special Hamiltonians—thereby obscuring the structural unity of the theory. This work develops a general, Hamiltonian-independent framework for both real- and imaginary-time Green-functions, valid for any combination of bosonic and fermionic degrees of freedom, including mixed systems such as interacting electron–phonon models. The key conceptual advance is the nature index η , which complements the conventional statistical sign ζ . While ζ encodes particle statistics, η distinguishes canonical quantization of systems with massless particles ( η = 0) from second quantization of systems with massive particles ( η = 1). This classification removes ambiguities in chemical-potential terms and unifies the treatment of photons, phonons, electrons, and their mixtures under a single notation. From this basis, we formulate generalized ensemble constructions, generalized Lehmann-representations, and a fluctuation-dissipation theorem valid for arbitrary Hamiltonians. Several results—such as the monotonicity of particle number in the chemical potential, an explicit formula for the quadratic contribution to the internal energy from the retarded Green-function, and the Hartree-contribution to the self-energy in electron–phonon systems—are either absent from, or only sparsely mentioned in, existing literature. We further introduce Green’s Chart, a visual and algebraic map that systematically connects all Green-function types through analytic continuation, Fourier-transforms, and Wick-rotations, enabling all to be generated from the Matsubara Green-function. Applications to both non-interacting and interacting models demonstrate the framework’s ability to compute densities of states, thermodynamic potentials, and spectral functions with a consistent formalism. The results provide a complete and reproducible toolkit for equilibrium many-body theory, consolidating scattered results into a unified structure suitable for both pedagogical and computational purposes.
Part I: Green's Functions in Mathematical Physics - Time-Independent Green's Functions / Time-dependent Green's Functions / Part II: One-Body Green's Functions - Physical Significance of One-Body Green's Functions / Green's Functions and Perturbation Theory / Green's Functions for Tight-Binding Hamiltonians / Single Impurity Scattering / Extension Theory for Lattice Green's Functions / Part III: Many-Body Green's Functions - Field Operators and Three Pictures / Definition and Properties of Many-Body Green's Functions / The Diagram Technique for Zero-Temperature Green's Functions / Definition and Properties of Matsubara Green's Functions / Diagram Technique for the Matsubara Green's Functions / Three Approximation Schemes of the Diagram Technique / Linear Response Theory / The Equation of Motion Technique for the Green's Functions / Magnetic Systems Described by Heisenberg Model / The Green's Functions for Boson Systems with Condensation / Superconductors with Weak Interaction Between Electrons / Nonequilibrium Green's Functions / Electronic Transport through a Mesoscopic Structure / Appendixes / Index.
Abstract Chapter 09 Nonequilibrium Green’s functions (NEGF), including coupled-correlated (C) single- and multi-particle Green’s functions, are defined as averages weighted with the time-development operator U(t0+τ,t0). Linear conductivity is exhibited as a two-particle equilibrium Green’s function (Kubo-type formulation). Admitting particle sources (S:η,η+) and non-conservation of number, the non-equilibrium multi-particle Green’s functions are constructed with numbers of creation and annihilation operators that may differ, and they may be derived as variational derivatives with respect to sources η,η+ of a generating functional eW=TrU(t0+τ,t0)CS/TrU(t0+τ,t0)C. (In the non-interacting case this yields the n-particle Green’s function as a permanent/determinant of single-particle Green’s functions.) These variational relations yield a symmetric set of multi-particle Green’s function equations. Cumulants and the Linked Cluster Theorem are discussed and the Random Phase Approximation (RPA) is derived variationally. Schwinger’s variational differential formulation of perturbation theories for the Green’s function, self-energy, vertex operator, and also shielded potential perturbation theory, are reviewed. The Langreth Algebra arises from analytic continuation of integration of products of Green’s functions in imaginary time to the real-time axis with time-ordering along the integration contour in the complex time plane. An account of the Generalized Kadanoff-Baym Ansatz is presented.