Superconductors are topologically ordered

TL;DR

This paper demonstrates that superconductors possess topological order characterized by global properties and fractionalization, unifying various observations through a topological $BF$ action and highlighting the role of electromagnetic fields.

Contribution

It reveals that superconducting states exhibit topological order described by a $BF$ topological field theory, unifying previous findings and extending to systems with boundaries and electromagnetic interactions.

Findings

Superconductors exhibit topological order with fractionalized excitations.

The $BF$ topological action predicts boundary states and ground state degeneracy.

Topological order described by $BF$ action applies to multiple physical systems.

Abstract

We revisit a venerable question: what is the nature of the ordering in a superconductor? We find that the answer is properly that the superconducting state exhibits topological order in the sense of Wen, i.e. that while it lacks a local order parameter, it is sensitive to the global topology of the underlying manifold and exhibits an associated fractionalization of quantum numbers. We show that this perspective unifies a number of previous observations on superconductors and their low lying excitations and that this complex can be elegantly summarized in a purely topological action of the ``$BF$'' type and its elementary quantization. On manifolds with boundaries, the $BF$ action correctly predicts non-chiral edge states, gapped in general, but crucial for fractionalization and establishing the ground state degeneracy. In all of this the role of the physical electromagnetic fields is central. We also observe that the $BF$ action describes the topological order in several other physically distinct systems thus providing an example of topological universality.