Superconductors are topologically ordered

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.

Effective field theories for topological insulators by functional bosonization

“…The former was discussed for the BF theory describing a BCS superconductor in Ref. 66, which has a Z 2 topological order. It would be interesting to see if one started from a U(1) × U(1) theory and implemented either symmetry breaking mechanism if one would generate a hydrodynamic field theory that is equivalent with what we constructed in this article by only using the U(1) charge symmetry.…”

Three-dimensional topological insulators and bosonization

“…• That of bosonic theories, also called hydrodynamic approach, dealing with topological gauge theories and their description of braiding relations and boundary excitations [12,[26][27][28][29][30][31].…”

Topological order and the deconfinement transition in the(2+1)-dimensional compact Abelian Higgs model

“…In the infinite system size limit the degenerate ground states may be characterized by having an even or odd number of flux quanta Φ 0 through the holes of the torus, giving a degeneracy of 2 2 = 4 since a torus has two holes. 32 The general case of arbitrary q on a manifold with genus g (i.e., which has g handles and 2g holes) naturally leads to q 2g degenerate ground states. For finite system size the degeneracy will be lifted by vortex tunneling processes, in which a vortex tunnels around a nontrivial path on the torus as illustrated in Fig.…”