Ward identities for strongly coupled Eliashberg theories

TL;DR

This paper derives Ward identities for strongly coupled Eliashberg theories, clarifying their compatibility with Migdal theorem and exploring their implications at quantum criticality, especially for charge and spin vertices.

Contribution

It provides a diagrammatic derivation of Ward identities in Eliashberg theories and discusses their limitations for spin vertices at quantum critical points.

Findings

Ward identities are compatible with Migdal theorem in Eliashberg theories.

Charge vertex Ward identity is derived for Fermi liquids and at quantum criticality.

Ward identity for spin vertex cannot be obtained within Eliashberg theory.

Abstract

We discuss Ward identities for strongly interacting fermion systems described by Eliashberg-type theories. We show that Ward identities are not in conflict with Migdal theorem. We derive diagrammatically Ward identity for a charge vertex in a Fermi liquid, and when a Fermi liquid is destroyed at quantum criticality. We argue that Ward identity for a spin vertex cannot be obtained within Eliashberg theory.