The modular S-matrix as order parameter for topological phase transitions

Abstract

We study topological phase transitions in discrete gauge theories in two spatial dimensions induced by the formation of a Bose condensate. We analyse a general class of euclidean lattice actions for these theories which contain one coupling constant for each conjugacy class of the gauge group. To probe the phase structure we use a complete set of open and closed anyonic string operators. The open strings allow one to determine the particle content of the condensate, whereas the closed strings enable us to determine the matrix elements of the modular $S$-matrix, also in the broken phase. From the measured broken $S$-matrix we may read off the sectors that split or get identified in the broken phase, as well as the sectors that are confined. In this sense the modular $S$-matrix can be employed as a matrix valued non-local order parameter from which the low-energy effective theories that occur in different regions of parameter space can be fully determined. To verify our predictions we studied a non-abelian anyon model based on the quaternion group $H=\overline{D_2}$ of order eight by Monte Carlo simulation. We probe part of the phase diagram for the pure gauge theory and find a variety of phases with magnetic condensates leading to various forms of (partial) confinement in complete agreement with the algebraic breaking analysis. Also the order of various transitions is established.

37 pages

References in corpus (7)

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• Bilayer quantum Hall phase transitions and the orbifold non-Abelian fractional quantum Hall states
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• Bulk Anyons as Edge Symmetries: Boundary Phase Diagrams of Topologically Ordered States
• Symmetry Enriched Phases via Pseudo Anyon Condensation
• Microscopic Study of the Halperin - Laughlin Interface through Matrix Product States
• Fermion Condensation and Gapped Domain Walls in Topological Orders
• Identifying Non-Abelian Topological Order through Minimal Entangled States
• A Unified Framework of Topological Phases with Symmetry
• Topological Characterization of Non-Abelian Moore-Read State using Density-Matrix Renormailzation Group
• Identifying non-Abelian topological ordered state and transition by momentum polarization
• Model wavefunctions for interfaces between lattice Laughlin states
• Transition of a ${\mathbb{Z}}_3$ topologically ordered phase to trivial and critical phases
• Chern-Simons theory encoded on a spin chain
• Model wavefunctions for an interface between lattice Laughlin and Moore-Read states
• Detecting transition between Abelian and non-Abelian topological orders through symmetric tensor networks
• Boson-Lattice Construction for Anyon Models