Shadows of Anyons

TL;DR

This paper uses tensor network methods to analyze the transfer matrix eigenstructure in topological phases, revealing how symmetry breaking relates to anyon excitations and phase transitions, thus advancing understanding of topological order and anyon condensation.

Contribution

It introduces a tensor network approach to connect transfer matrix symmetry properties with anyon excitations and phase transitions in topological quantum matter.

Findings

Topological order requires specific symmetry breaking in the transfer matrix fixed point.

Anyons correspond to domain wall excitations in the transfer matrix.

Phase transitions involve changes in symmetry related to anyon condensation or confinement.

Abstract

The eigenvalue structure of the quantum transfer matrix is known to encode essential information about the elementary excitations. Here we study transfer matrices of quantum states in a topological phase using the tensor network formalism. We demonstrate that topological quantum order requires a particular type of `symmetry breaking' for the fixed point subspace of the transfer matrix, and relate physical anyon excitations to domain wall excitations at the level of the transfer matrix. A topological phase transition to a trivial phase triggers a change in the fixed point subspace to either a larger or smaller symmetry and we explain how this relates to a condensation or confinement of the corresponding anyon sectors. The tensor network formalism enables us to determine the structure of the topological sectors in two-dimensional gapped phases very efficiently, therefore opening novel avenues for studying fundamental open questions related to anyon condensation.