Interaction effects on 3D topological superconductors: surface topological order from vortex condensation, the 16 fold way and fermionic Kramers doublets

TL;DR

This paper investigates how interactions modify the classification of 3D topological superconductors with time reversal symmetry, revealing a reduction from an integer classification to a finite cyclic group and exploring related surface topological orders.

Contribution

It explicitly derives surface topological orders for even-indexed phases using vortex condensation and connects these results to known classifications in 1D topological superconductors.

Findings

Interactions reduce the free fermion classification from Z to Z16 for 3D topological superconductors.

Explicit derivation of surface topological orders for even-index phases using vortex condensation.

Identification of fermionic Kramers doublets with $T^2=\pm i$ in surface topological orders.

Abstract

Three dimensional topological superconductors with time reversal symmetry (class DIII) are indexed by an integer $\nu$, the number of surface Majorana cones, according to the free fermion classification. The superfluid B phase of He$^3$ realizes the $\nu=1$ phase. Recently, it has been argued that this classification is reduced in the presence of interactions to Z$_{16}$. This was argued from the symmetry respecting surface topological orders of these states, which provide a non-perturbative definition of the bulk topological phase. Here, we verify this conclusion by focusing on the even index case, $\nu=2m$, where a vortex condensation approach can be used to explicitly derive the surface topological orders. We show a direct relation to the well known result on one dimensional topological superconductors (class BDI), where interactions reduce the free fermion classification from Z down to Z$_8$. Finally, we discuss in detail the fermionic analog of Kramers time reversal symmetry, which allows semions of some surface topological orders to transform as $T^2=\pm i$.