The effect of interactions on 2D fermionic symmetry-protected topological phases with Z2 symmetry

Abstract

We study the effect of interactions on 2D fermionic symmetry-protected topological (SPT) phases using the recently proposed braiding statistics approach. We focus on a simple class of examples: superconductors with a Z2 Ising symmetry. Although these systems are classified by Z in the noninteracting limit, our results suggest that the classification collapses to Z8 in the presence of interactions -- consistent with previous work that analyzed the stability of the edge. Specifically, we show that there are at least 8 different types of Ising superconductors that cannot be adiabatically connected to one another, even in the presence of strong interactions. In addition, we prove that each of the 7 nontrivial superconductors have protected edge modes.

Figures (1)

• Figure 1: (a) We consider a thought experiment in which we create two $Z^\downarrow_2$ fluxes in the bulk and then move them along a path $\beta$ to points $a,b$ at the edge. (b) We argue that the two fluxes can be annihilated at the boundary by applying local operators $U_a$, $U_b$. (c) We define $\mathbb{W}_\beta$ to be an operator which describes a process in which the fluxes are created in the bulk, brought to the edge, and then annihilated. (d) To obtain a contradiction, we consider two paths $\beta, \gamma$ that intersect one another, and we investigate the commutation algebra of the corresponding operators $\mathbb{W}_\beta$,$\mathbb{W}_\gamma$.