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Physical review. B, Condensed matter

Exact formula for nondiagonal Green’s functions in condensed-matter physics

Tadashi Toyoda, Yasushi Takahashi, Chao Zhang

Published Dec 15, 1994• 7 citations
Publisher

Abstract

An exact formula for the nondiagonal part of the one-particle temperature Green's function is obtained within the framework of the nonperturbative canonical formulation of the finite-temperature generalized Ward-Takahashi relation. It is shown that the Landau-level Green's function for two-dimensional electrons has a nonvanishing nondiagonal part if the electron-electron interaction is included.

DOI
10.1103/physrevb.50.18640

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An exact formula for the nondiagonal part of the one-particle temperature Green's function is obtained within the framework of the nonperturbative canonical formulation of the finite-temperature generalized Ward-Takahashi relation. It is shown that the Landau-level Green's function for two-dimensional electrons has a nonvanishing nondiagonal part if the electron-electron interaction is included.
Without any approximation, exact relations for the spin-density correlation functions in an interacting electron gas under an arbitrary nonuniform external magnetic field are obtained by making use of the nonperturbative canonical formulation of quantum many-body theory [T. Toyoda, Ann. Phys. (N.Y.) 173, 226 (1987)]. The obtained results contain fff-sum rules, a finite-temperature generalized Ward-Takahashi relation, as well as a finite-temperature version of the Nambu-Goldstone theorem with respect to the spontaneous symmetry breaking of the spin-rotational symmetry.
We derive Ward-Takahashi identities for correlated Bose-Einstein condensates based on the expressions of the first-order variations (δΨ,δG)(δΨ,δG)(δΨ,δG) due to perturbations obtained in the preceding paper [T. Kita, J. Phys. Soc. Jpn. 90\bf 9090, 024001 (2021)] for the condensate wave function ΨΨΨ and Green's function GGG. They enable us to obtain several exact results on the density and current correlation functions Kνν′K_{νν'}^{}Kνν′​, and also express Kνν′K_{νν'}^{}Kνν′​ in terms of low-energy Green's functions and vertices. The latter expressions open up the possibility of constructing theory of superfluid Bose liquids in the same way as that for fermions at low temperatures. The vertices are found to have different limits depending on which of frequency ωωω and wavenumber qqq is set equal to zero first.
A nonperturbative argument is used to derive a one-electron Hamiltonian whose eigenfunctions are in one-to-one correspondence with electron or hole quasiparticles of an N-electron system. Nonlocal exchange is included exactly, and self-energy terms due to electronic correlation are included at the level of approximation of a given N-electron wave function. When parametrized by quasiparticle occupation numbers, this one-electron Hamiltonian is the functional derivative of an exact N-electron energy functional, as in the Landau theory of interacting fermion systems. The present nonperturbative derivation extends the Landau theory to ordered or localized systems for which many-body perturbation theory is not directly applicable. Analysis of the Bardeen-Cooper-Schrieffer (BCS) reduced Hamiltonian for superconductors leads to a temperature-dependent energy gap and a transition temperature in good agreement with the BCS theory, in a formalism that conserves the number of electrons.
The Fermi-liquid transport theory of electrical conductivity formulated by Eliashberg [Zh. Eksp. Teor. Fiz. 41, 1241 (1961)] is further developed, using the analytic continuation of the finite-temperature Ward-Takahashi relations obtained by the present author [Ann. Phys. (N.Y.) 173, 226 (1987)] to maintain the theoretical consistency of approximation schemes for solving the Bethe-Salpeter equation. In particular, the finite-temperature Ward-Takahashi relations corresponding to the local conservation of the electron number density and to the conservation of the electron momentum are considered. A general expression for the electrical conductivity in terms of the proper vertex part is also obtained. In the formula, cancellation of the quasiparticle renormalization factors at the Fermi surface is explicitly shown in a general way without assuming a particular model. In the case that the electrons are scattered by phonons, the formula gives the Gr\"uneisen-Bloch formula for the on-shell phonons. For the impurity resistivity, the theory of Edwards [Philos. Mag. 3, 1020 (1958)] is developed to finite temperature by introducing a hypothetical boson field that replaces the impurity averaging.
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.