Effect of interactions on two-dimensional fermionic symmetry-protected topological phases with $Z_2$ symmetry

We study the effect of interactions on two-dimensional fermionic symmetry-protected topological (SPT) phases using the recently proposed braiding statistics approach. We focus on a simple class of examples: superconductors with a $Z_2$ Ising symmetry. Although these systems are classified by $\mathbb{Z}$ in the noninteracting limit, our results suggest that the classification collapses to ${\mathbb{Z}}_8$ in the presence of interactions—consistent with previous work that analyzed the stability of the edge. Specifically, we show that there are at least eight 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 seven nontrivial superconductors have protected edge modes.

Figure 1

(a) We consider a thought experiment in which we create two $Z_2^{\downarrow }$ 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 ${\mathbb{W}}_{\beta }$,${\mathbb{W}}_{\gamma }$.