# Anyons in the $\pi$-flux phase of fermionic matter coupled to a $\mathbb{Z}_2$-gauge field

## TL;DR

This paper proves that a system of fermions coupled to a $bZ_2$ gauge field has a topologically ordered, gapped ground state with anyonic excitations, using reflection positivity and flux insertion techniques.

## Contribution

It demonstrates the topological order and anyonic braiding properties in a fermionic lattice system coupled to a $bZ_2$ gauge field, with rigorous proofs of flux sector and monopole mass.

## Key findings

- Ground state in uniform $bZ_2$ flux sector
- Monopoles are massive and gapped
- Braiding statistics match those of the toric code

## Abstract

We consider a system of weakly interacting spinful lattice fermions coupled to a dynamical $\mathbb{Z}_2$ gauge field. Using reflection positivity, we prove that the ground state lies in the sector of a uniform $\pi$-flux per plaquette and that the monopoles are massive. In the presence of a staggered mass for the fermions, this yields a fully gapped, four-dimensional ground state space on large tori. It is topologically ordered. By considering adiabatic $\pi$-flux insertion, we construct dressed monopole excitations, show that their self-braiding is proportional to the Hall conductance and hence vanishes, and prove that their braiding with the fermionic excitations is that of the toric code.

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Source: https://tomesphere.com/paper/2508.21502