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

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.

9+16 pages, 3+8 figures

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• Error-correcting codes for fermionic quantum simulation
• Classification and Quantification of Entanglement Through Wedge Product and Geometry
• Non-Abelian Anyons in Periodically Driven Abelian Spin Liquids
• Anyon polarons as a window into competing phases of the Kitaev honeycomb model under a Zeeman field
• Wegner's Ising gauge spins versus Kitaev's Majorana partons: Mapping and application to anisotropic confinement in spin-orbital liquids
• Jordan-Wigner composite-fermion liquids in 2D quantum spin-ice