Local Ward identities for collective excitations in fermionic systems with spontaneously broken symmetries

TL;DR

This paper derives Ward identities linking gauge field responses to order parameter susceptibilities in fermionic systems with spontaneously broken gauge symmetry, providing insights into collective excitations in superconductors, antiferromagnets, and spiral magnets.

Contribution

It introduces a general derivation of Ward identities for broken gauge symmetries in fermionic systems and applies them to specific magnetic and superconducting cases, including explicit calculations.

Findings

Ward identities relate response functions to order parameter susceptibilities.

Explicit expressions for spin stiffnesses and wave velocities in spiral magnets.

Validation of Ward identities within the random phase approximation.

Abstract

We derive Ward identities for fermionic systems exhibiting a gauge symmetry that gets globally broken. In particular, we focus on the relation that connects the gauge field response functions to the transverse susceptibilities of the order parameter. We find that the long-wavelength and zero energy limit of the former are related to the coefficients of a low-energy expansion of the latter. We examine three physical cases: the superconductor, the N\'eel antiferromagnet and the spiral magnet. In the case of a metallic spiral magnet that completely breaks the SU(2) spin symmetry we explicitly show that the Ward identities are fulfilled within the random phase approximation. We subsequently derive microscopic expressions for the spin stiffnesses and spin wave velocities, which can be plugged into low energy models to study the effect of long-wavelength bosonic fluctuations on top of mean-field solutions.