Gauge Invariance and Symmetry Breaking by Topology and Energy Gap

TL;DR

This paper explores how topology and energy gaps influence gauge invariance and symmetry breaking in quantum systems, revealing mechanisms distinct from Goldstone and Higgs phenomena, with examples like particles on a circle and electrons in crystals.

Contribution

It introduces a novel framework linking topological invariants to spontaneous symmetry breaking in quantum systems, differing from traditional Goldstone and Higgs mechanisms.

Findings

Symmetry breaking is tied to the fundamental group of the configuration space.

Energy gaps can lead to spontaneous symmetry breaking without Goldstone bosons.

Topological invariants can induce gauge symmetry breaking in quantum models.

Abstract

For the description of observables and states of a quantum system, it may be convenient to use a canonical Weyl algebra of which only a subalgebra $\mathcal A$, with a non-trivial center $\mathcal Z$, describes observables, the other Weyl operators playing the role of intertwiners between inequivalent representations of $\mathcal A$. In particular, this gives rise to a gauge symmetry described by the action of $\mathcal Z$. A distinguished case is when the center of the observables arises from the fundamental group of the manifold of the positions of the quantum system. Symmetries which do not commute with the topological invariants represented by elements of $\mathcal Z$ are then spontaneously broken in each irreducible representation of the observable algebra, compatibly with an energy gap; such a breaking exhibits a mechanism radically different from Goldstone and Higgs mechanisms. This is clearly displayed by the quantum particle on a circle, the Bloch electron and the two body problem.