Electric-magnetic duality of lattice systems with topological order

TL;DR

This paper explores the electromagnetic duality in lattice models with topological order, extending Kitaev's models to Hopf algebra-based systems, and interprets duality as an invertible domain wall with applications in topology measurement.

Contribution

It introduces a new class of topological models based on Hopf algebras, extending Kitaev's quantum double models, and interprets EM duality as an invertible domain wall.

Findings

EM duality defined for all Kitaev models

Extended models based on Hopf algebras include Levin-Wen string-nets

EM duality can be viewed as an invertible domain wall

Abstract

We investigate the duality structure of quantum lattice systems with topological order, a collective order also appearing in fractional quantum Hall systems. We define electromagnetic (EM) duality for all of Kitaev's quantum double models based on discrete gauge theories with Abelian and non-Abelian groups, and identify its natural habitat as a new class of topological models based on Hopf algebras. We interpret these as extended string-net models, whereupon Levin and Wen's string-nets, which describe all intrinsic topological orders on the lattice with parity and time-reversal invariance, arise as magnetic and electric projections of the extended models. We conjecture that all string-net models can be extended in an analogous way, using more general algebraic and tensor-categorical structures, such that EM duality continues to hold. We also identify this EM duality with an invertible domain wall. Physical applications include topology measurements in the form of pairs of dual tensor networks.