Berry Phases of Vison Transport in $\mathbb{Z}_2$ Topologically Ordered States from Exact Fermion-Flux Lattice Dualities

TL;DR

This paper introduces an exact duality mapping for 2D spin-1/2 systems to fermions and bosons, enabling explicit calculation of Berry phases for flux transport in various topologically ordered states, including non-abelian cases.

Contribution

It develops a novel duality that captures charge and flux dynamics without local conservation constraints, facilitating computation of vison Berry phases in complex topologically ordered states.

Findings

Numerical conditions for Majorana flux phases in Ising topological order.

Explicit calculation of vison Berry phases in symmetry enriched topological states.

Reduction of complex phase computations to overlaps of free-fermion ground states.

Abstract

We develop an exact map of all states and operators from 2D lattices of spins-$1/2$ into lattices of fermions and bosons with mutual semionic statistical interaction that goes beyond previous dualities of $\mathbb{Z}_2$ lattice gauge theories because it does not rely on imposing local conservation laws and captures the motion of ``charges'' and ``fluxes'' on equal footing. This map allows to explicitly compute the Berry phases for the transport of fluxes in a large class of symmetry enriched topologically ordered states with emergent $\mathbb{Z}_2$ gauge fields that includes chiral, non-chiral, abelian or non-abelian, that can be perturbatively connected to models where the visons are static and the emergent fermionic spinons have a non-interacting dispersion. The numerical complexity of computing such vison phases reduces therefore to computing overlaps of ground states of free-fermion Hamiltonians. Among other results, we establish numerically the conditions under which the Majorana-carrying flux excitation in Ising-Topologically-Ordered states enriched by translations acquires $0$ or $\pi$ phase when moving around a single plaquette.