# The nature of symmetry breaking in the superconducting ground state

## TL;DR

This paper introduces a gauge-invariant criterion for detecting symmetry breaking in superconductors, clarifying the role of custodial symmetry and providing numerical analysis of phase boundaries in lattice models.

## Contribution

It proposes a new gauge-invariant method to identify symmetry breaking, addressing ambiguities of traditional order parameters in superconductors.

## Key findings

- Spontaneous custodial symmetry breaking is necessary for global gauge symmetry breaking.
- The criterion accurately identifies phase boundaries in lattice Ginzburg-Landau models.
- Numerical results confirm the theoretical relationship between custodial and gauge symmetry breaking.

## Abstract

The order parameters which are thought to detect U(1) gauge symmetry breaking in a superconductor are both non-local and gauge dependent. For that reason they are also ambiguous as a guide to phase structure. We point out that a global subgroup of the local U(1) gauge symmetry may be regarded, in analogy to non-abelian theories, as a "custodial" symmetry affecting the matter field alone, and construct, along the lines of our previous work, a new gauge-invariant criterion for breaking symmetries of this kind. It is shown that spontaneous breaking of custodial symmetry is a necessary condition for the existence of spontaneous symmetry breaking of a global subgroup of the (abelian or non-abelian) gauge group in any given gauge, and a sufficient condition for the existence of spontaneous breaking of a global subgroup of the gauge group in some gauge. As an illustration we compute numerically, in the lattice version of the Ginzburg-Landau model, the phase boundaries of the theory and the order parameters associated with various symmetries in each phase.