Berry phases of vison transport in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi mathvariant="double-struck">Z</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math> topologically ordered states from exact fermion-flux lattice dualities
Chuan Chen, Peng Rao, Inti Sodemann
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 ${\mathbb{Z}}_{2}$ 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, 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 $\ensuremath{\pi}$ phase when moving around a single plaquette.