Pseudo-spontaneous $U(1)$ Symmetry Breaking in Hydrodynamics and Holography

TL;DR

This paper develops a hydrodynamic theory for systems with pseudo-spontaneously broken $U(1)$ symmetry, compares it with holographic models, and finds universal relations linking phase relaxation, Goldstone mass, and diffusivity.

Contribution

It introduces a hydrodynamic framework for pseudo-spontaneous $U(1)$ breaking and demonstrates its agreement with holographic models, revealing universal relations in phase relaxation phenomena.

Findings

Hydrodynamic dispersion relations match holographic quasi-normal modes.

Phase relaxation occurs only due to explicit Goldstone shift symmetry breaking.

Universal relation between phase relaxation rate, Goldstone mass, and diffusivity is confirmed.

Abstract

We investigate the low-energy dynamics of systems with pseudo-spontaneously broken $U(1)$ symmetry and Goldstone phase relaxation. We construct a hydrodynamic framework which is able to capture these, in principle independent, effects. We consider two generalisations of the standard holographic superfluid model by adding an explicit breaking of the $U(1)$ symmetry by either sourcing the charged bulk scalar or by introducing an explicit mass term for the bulk gauge field. We find agreement between the hydrodynamic dispersion relations and the quasi-normal modes of both holographic models. We verify that phase relaxation arises only due to the breaking of the inherent Goldstone shift symmetry. The interplay of a weak explicit breaking of the $U(1)$ and phase relaxation renders the DC electric conductivity finite but does not result in a Drude-like peak. In this scenario we show the validity of a universal relation, found in the context of translational symmetry breaking, between the phase relaxation rate, the mass of the pseudo-Goldstone and the Goldstone diffusivity.