Superconductors are topologically ordered

TL;DR

This work argues that gapped superconductors are examples of topological order, lacking a local order parameter and describable by a BF topological field theory that encodes braiding between charges and vortices. It derives the BF action from the abelian Higgs model, demonstrates a robust ground-state degeneracy on nontrivial manifolds, and analyzes edge states arising at boundaries, showing how bulk topological data constrain boundary dynamics. By connecting superconductors to other systems such as Z2 lattice gauge theories and RVB states, the paper establishes a broader topological universality class and clarifies how non-topological terms restore electrodynamics (plasmons and London screening). It also outlines how instantons and boundary conditions modulate charge conservation and edge physics, and sketches avenues for non-Abelian generalizations and gapless extensions, with implications for understanding unconventional superconductors and related quantum phases.

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.

Figures (10)

• Figure 2: Topological interactions in a superconductor: quasiparticles encircling vortices ($d=2$) or threading vortex loops ($d=3$) pick up a phase $\pi$ at an arbitrary distance.
• Figure 3: A vortex tunnelling process inserting a unit of magnetic flux inside the torus. In this visualization it also leaves a flux loop outside, but that is invisible to the electrons on the surface. This process connects ground states labelled by opposite values of the Wilson loop $e^{i \oint_C \vec{a} \cdot {\vec{d}l}} \equiv e^{i e \Phi_M}$ where $\Phi_M$ is the magnetic flux threading $C$.
• Figure 4: A quasiparticle-pair tunnelling process changing the value of the modular electric flux, $e^{i \oint_C \vec{b} \cdot {\vec{d}l}} \equiv e^{i \frac{\pi}{e} \Phi_E}$ where $\Phi_E$ is the surface electric flux crossing $C$.
• Figure 5: Two of the four ground states of the $Z_{2}\ $ lattice gauge theory at zero coupling, on the torus. They differ by the insertion of a $Z_{2}\ $ vortex (vison) through one of the holes of the torus---which is implemented by changing the sign on a string of bonds as shown. The pair of states thus differ in the sign of the Wilson loop $\Pi_C \sigma^z$. The remaining two states differ by vison insertion in the other hole.
• Figure 6: Geometry for establishing the $\pi$ linking phase between Wilson loops on the direct and dual lattices, $W_{\Gamma_1}$ and $W_{\Gamma_2}$ in Eqn. (\ref{['lloops']})