Is electromagnetic gauge invariance spontaneously violated in superconductors?

TL;DR

This paper provides a pedagogical review clarifying that global U(1) symmetry, not gauge symmetry, is spontaneously broken in superconductors, and demonstrates that the BCS wave function is gauge invariant, explaining key superconducting phenomena.

Contribution

It offers a detailed, clear explanation of the symmetry breaking and gauge invariance in superconductivity, correcting misconceptions and deriving the effective field theory from fundamental assumptions.

Findings

Global U(1) symmetry, not gauge symmetry, is spontaneously broken in superconductors.

The BCS wave function is fully gauge invariant, contrary to some claims.

Superconducting phenomena follow from a charged order parameter field.

Abstract

We aim to give a pedagogical introduction to those elementary aspects of superconductivity which are not treated in the classic textbooks. In particular, we emphasize that global U(1) phase rotation symmetry, and not gauge symmetry, is spontaneously violated, and show that the BCS wave function is, contrary to claims in the literature, fully gauge invariant. We discuss the nature of the order parameter, the physical origin of the many degenerate states, and the relation between formulations of superconductivity with fixed particle numbers vs. well defined phases. We motivate and to some extend derive the effective field theory at low temperatures, explore symmetries and conservation laws, and justify the classical nature of the theory. Most importantly, we show that the entire phenomenology of superconductivity essentially follows from the single assumption of a charged order parameter field. This phenomenology includes Anderson's characteristic equations of superfluidity, electric and magnetic screening, the Bernoulli Hall effect, the balance of the Lorentz force, as well as the quantum effects, in which Planck's constant manifests itself through the compactness of the U(1) phase field. The latter effects include flux quantization, phase slippage, and the Josephson effect.