The boundaries of 2+1D fermionic topological orders

TL;DR

This paper develops a systematic method to determine gapped boundaries of 2+1D abelian fermionic topological orders by constructing bosonic extensions and using $K$-matrix formalisms, with applications to Laughlin states.

Contribution

It introduces a bosonic extension approach to analyze fermionic topological orders and provides an explicit algorithm for boundary partition functions and excitations.

Findings

Constructed a correspondence between abelian FTOs and fermion-condensed $Z_2$ topological orders.

Derived boundary properties for Laughlin states with square $m$, including their fusion rings.

Generalized the method to non-abelian fermionic topological orders.

Abstract

$2+1$D bosonic topological orders can be characterized by the $S,T$ matrices that encode the statistics of topological excitations. In particular, the $S,T$ matrices can be used to systematically obtain the gapped boundaries of bosonic topological orders. Such an approach, however, does not naively apply to fermionic topological orders (FTOs). In this work, we propose a systematic approach to obtain the gapped boundaries of $2+1$D abelian FTOs. The main trick is to construct a bosonic extension in which the fermionic excitation is "condensed" to form the associated FTOs. Here we choose the parent bosonic topological order to be the $\mathbb{Z}_2$ topological order, which indeed has a fermionic excitation. Such a construction allows us to find an explicit correspondence between abelian FTOs (described by odd $K$-matrix $K_F$) and the "fermion-" condensed $\mathbb{Z}_2$ topological orders (described by even $K$-matrix $K_B$). This provides a systematic algorithm to obtain the modular covariant boundary partition functions as well as the boundary topological excitations of abelian FTOs. For example, the $\nu=1-\frac{1}{m}$ Laughlin's states have exactly one type of gapped boundary when $m$ is a square, whose boundary excitations form a $\mathbb{Z}_{2}\times\mathbb{Z}_{\sqrt{m}}$ fusion ring. Our approach can be easily generalized to obtain gapped and gapless boundaries of non-abelian fermionic topological orders.