Numerical detection of symmetry enriched topological phases with space group symmetry

TL;DR

This paper introduces a numerical method to detect symmetry-enriched topological phases, specifically crystal momentum fractionalization, using PEPS and transfer matrix spectrum analysis, demonstrated on a modified toric code model.

Contribution

The paper presents a novel numerical approach to identify symmetry fractionalization in topological phases via transfer matrix spectrum analysis of PEPS.

Findings

Enhanced periodicity in momentum spectrum indicates anticommutation of translation operators.

Method successfully detects symmetry fractionalization in a perturbed toric code model.

Transfer matrix spectrum analysis reveals quantum number fractionalization in topological phases.

Abstract

Topologically ordered phases of matter, in particular so-called symmetry enriched topological (SET) phases, can exhibit quantum number fractionalization in the presence of global symmetry. In Z_2 topologically ordered states in two dimensions, fundamental translations T_x and T_y acting on anyons can either commute or anticommute. This property, crystal momentum fractionalization, can be seen in a periodicity of the excited-state spectrum in the Brillouin zone. We present a numerical method to detect the presence of this form of symmetry enrichment given a projected entangled pair state (PEPS); we study the minima of spectrum of correlation lengths of the transfer matrix for a cylinder. As a benchmark, we demonstrate our method using a modified toric code model with perturbation. An enhanced periodicity in momentum clearly reveals the anticommutation relation {T_x,T_y}=0$ for the corresponding quasiparticles in the system.