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

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.

Figure 1

Four Dirac fermions ${\psi }_{i\alpha }$ arising from the two pairs of nodes (connected by dashed lines) at the Fermi surface of a $d_{x^2-y^2}$ superconductor.

Figure 2

Topological phase diagram of a deformed $d_{x^2-y^2}$ superconductor as a function of masses $m_i$ of the Dirac modes ${\psi }_i$. The origin represents the $d_{x^2-y^2}$ superconductor.

Figure 3

Integer-valued $l$ vectors of 1, $e, m, \epsilon$ excitations for the $s$-wave and $d+id$ superconductor.

Figure 4

Diagrammatic representation of the 16-fold way. The Abelian states considered in this paper are represented by thick lines and the other Abelian states by thin lines. Dashed lines indicate the non-Abelian states. For each state, the angle $\theta$ equals the exchange angle ${\theta }_e={\theta }_m$.

Figure 5

A chiral superconductor with chirality $k\in 2\mathbb{Z}$ hosts $2k$ Majorana modes at the edge. As indicated by the $(\uparrow \downarrow )$ arrows, each mode is doubly degenerate due to spin symmetry and modes with the same color are related by particle-hole symmetry. Thus, the modes within a quartet have the same velocity $v_i$, while generically different quartets will have different velocities.

Figure 6

Feynman diagram for the kernel ${\mathrm{\Gamma }}^{\mu \nu }\left(p\right)$: the bold wavy line denotes the renormalized photon propagator $i{D}_{\gamma \delta }\left(p\right)$; the red cross represents the vertex originating from the last term in Eq. (47).