Topological order, symmetry, and Hall response of two-dimensional spin-singlet superconductors

TL;DR

This paper develops a unified topological field theory framework for 2D spin-singlet superconductors, analyzing their topological order, quasiparticle statistics, and Hall responses, including edge physics and symmetry considerations.

Contribution

It derives Chern-Simons theories for various singlet superconductors, linking their topological properties to Kitaev's classification and analyzing their Hall responses and edge states.

Findings

Chiral d+id superconductor has spin Hall coefficient ν_s=2.

Magnetic flux Hall response vanishes in these systems.

Ground state of chiral superconductors lacks spontaneous magnetic field.

Abstract

Fully gapped two-dimensional superconductors coupled to dynamical electromagnetism are known to exhibit topological order. In this work, we develop a unified low-energy description for spin-singlet paired states by deriving topological Chern-Simons field theories for $s$-wave, $d+id$, and chiral higher even-wave superconductors. These theories capture the quantum statistics and fusion rules of Bogoliubov quasiparticles and vortices and incorporate global continuous symmetries - specifically, spin rotation and conservation of magnetic flux - present in all singlet superconductors. For all such systems, we compute the Hall response for these symmetries and investigate the physics at the edge. In particular, the weakly-coupled phase of a chiral $d+id$ chiral state has a spin Hall coefficient $\nu_s=2$ and a vanishing Hall response for the magnetic flux symmetry. We argue that the latter is a generic result for two-dimensional superconductors with gapped photons, thereby demonstrating the absence of a spontaneous magnetic field in the ground state of chiral superconductors. It is also shown that the Chern-Simons theories of chiral spin-singlet superconductors derived here fall into Kitaev's 16-fold classification of topological superconductors.