Numerical identification and gapped boundaries of abelian fermionic topological order

TL;DR

This paper investigates Abelian fermionic topological orders in 2D, focusing on their differences from bosonic phases, and proposes a numerical scheme for their identification without relying on fermion number conservation.

Contribution

It introduces a minimal scheme for numerically identifying Abelian fermionic topological orders and analyzes the conditions for gapped boundaries in these phases.

Findings

Gauging fermion parity reveals a simple structure useful for identification.

A gapped boundary in the gauged theory implies a gapped boundary in the ungauged fermion theory.

Subtleties of the momentum polarization technique are discussed.

Abstract

In this work we consider general fermion systems in two spatial dimensions, both with and without charge conservation symmetry, which realize a nontrivial fermionic topological order with only Abelian anyons. We address the question of precisely how these quantum phases differ from their bosonic counterparts, both in terms of their edge physics and in the way one would identify them in numerics. As in previous works, we answer these questions by studying the theory obtained after gauging the global fermion parity symmetry, which turns out to have a special and simple structure. Using this structure, a minimal scheme is outlined for how to numerically identify a general Abelian fermionic topological order, without making use of fermion number conservation. Along the way, some subtleties of the momentum polarization technique are discussed. Regarding the edge physics, it is shown that the gauged theory can have a (bosonic) gapped boundary to the vacuum if and only if the ungauged fermion theory has a gapped boundary as well.