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Ward Identities for Interacting Electronic Systems

H. T. Nieh, Ping Sheng, Xiaobing Wang

Published Jan 1, 1998• 2 citations
OpenAlex

Abstract

A Ward-Takahashi identity, as a consequence of gauge invariance and in a form that relates self-energy to the two-particle Bethe-Salpeter scattering kernel, was first derived by Vollhardt and Wölfle for a system of independent particles moving in a random medium. This is generalized to a class of interacting electronic systems in materials with or without random impurities, following a procedure previously used for classical waves transport in disordered media. This class of systems also possesses other symmetry properties such as invariance under time translations and local spin rotations, which imply local conservation laws for energy and spin current. They imply additional Vollhardt-Wölfle type identities. We present non-perturbative derivations of these identities, and consider the constraints they impose on the relationship between the self-energy and the two-particle scattering kernel. PACS numbers: 11.30.-j, 03.80.+r, 72.10.Bg 1 Ward identities are statements of local conservation laws, and reflect the corresponding

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A Ward-Takahashi identity, as a consequence of gauge invariance and in a form that relates self-energy to the two-particle Bethe-Salpeter scattering kernel, was first derived by Vollhardt and Wölfle for a system of independent particles moving in a random medium. This is generalized to a class of interacting electronic systems in materials with or without random impurities, following a procedure previously used for classical waves transport in disordered media. This class of systems also possesses other symmetry properties such as invariance under time translations and local spin rotations, which imply local conservation laws for energy and spin current. They imply additional Vollhardt-Wölfle type identities. We present non-perturbative derivations of these identities, and consider the constraints they impose on the relationship between the self-energy and the two-particle scattering kernel. PACS numbers: 11.30.-j, 03.80.+r, 72.10.Bg 1 Ward identities are statements of local conservation laws, and reflect the corresponding
In a Lagrange formulation of nonrelativistic quantum mechanics gauge invariance corresponds to a local U(1) symmetry of the Lagrangian density. We review how this gauge symmetry yields the electromagnetic interaction of the free and the locally interacting N-body system and why nonlocal interactions require additional exchange contributions. A nonrelativistic Ward-Takahashi identity is then derived. This relates electromagnetic operators to the full propagator of the particles in question, which may be off their energy shell.
We derive the Ward-Takahashi identity and establish the gauge-invariant response theory for open quantum systems described by Lindbladians to show that particle-number conservation is not necessary to satisfy gauge invariance. We construct an observable which can be used to test the gauge invariance in the absence of particle-number conservation. We derive the low-energy collective modes that emerge as a consequence of gauge invariance in open quantum systems, and find that two-body loss induces diffusive modes in dissipative Bardeen-Cooper-Schrieffer (BCS) superconductivity. Possible experimental situations for testing gauge invariance in open quantum systems are also discussed.
Using functional methods and the exact renormalization group we derive Ward identities for the Anderson impurity model. In particular, we present a non-perturbative proof of the Yamada–Yosida identities relating certain coefficients in the low-energy expansion of the self-energy to the thermodynamic particle number and spin susceptibilities of the impurity. Our proof underlines the relation of the Yamada–Yosida identities to the U (1) × U (1) symmetry associated with the particle number and spin conservation in a magnetic field.
A nonequilibrium Ward identity (NE WI) connecting the scalar transport vertex correction with one-particle self-energy is derived using the global U(1)\mathrm{U}(1)U(1) symmetry of the fermion nonequilibrium Green's functions (NGF). The nonperturbative derivation does not depend on the details of the many-body system. A renormalized multiplicative composition rule for the NGF, reflecting time coherence, is obtained and related to the NE WI. Applications involve (i) testing the consistency of approximations shown in the example of a self-consistent Born approximation for disorder scattering, and (ii) in the general quantum transport theory, the formalism permits one to assess routes to generalized master equations, in particular those based on the generalized Kadanoff-Baym ansatz.
Ward-Takahashi relations for spin symmetry SU(2)S\mathrm{SU}{(2)}_{\mathrm{S}}SU(2)S​ and pseudospin symmetry SU(2)C\mathrm{SU}{(2)}_{\mathrm{C}}SU(2)C​ are derived for the simple Hubbard model in arbitrary dimensions. For this purpose we generally discuss local SO(4) symmetry in terms of Clifford algebra, and employ a gauge transformed form of the original Hubbard Hamiltonian, which makes the Ward-Takahashi relations much simpler. Derived Ward-Takahashi relations are rewritten as the relations between self-energy and vertex part to discuss spontaneously broken symmetry and Fermi liquid property of the system for zero temperature magnetism.
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.