Berry phases of vison transport in Z_{2} topologically ordered states from exact fermion-flux lattice dualities

Abstract

We develop an exact map of all states and operators from two-dimensional lattices of spins-1/2 into lattices of fermions and bosons with mutual semionic statistical interaction that goes beyond previous dualities of 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 Z_{2} gauge fields that includes chiral, nonchiral, Abelian or non-Abelian, that can be perturbatively connected to models where the visons are static and the emergent fermionic spinons have a noninteracting 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 the 0 or π phase when moving around a single plaquette.