The modularS-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, in both the unbroken and broken phases. 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 nonlocal 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 =D 2 of the order of 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.

Fermion condensation and gapped domain walls in topological orders

“…In these proposals, a collection of anyons may form a boson and then condense, in a fashion similar to Cooper pair condensation. More recently, Bais et al proposed a set of general and practical rules of condensing self-bosons that have nontrivial braiding statistics with some other anyons in a bosonic topological order (bTO) [12][13][14][15]. After their condensation, the condensed self-bosons result in a new vacuum, and the original bTO undergoes a phase transition to a new bTO.…”

Diagrammatics for Bose condensation in anyon theories

“…In previous work 7 , we had already identified such nontrivial contributions to the S-matrix in a lattice model and dubbed them vacuum exchange diagrams.…”

“…First, one could start from a lattice model and add perturbing terms to its Hamiltonian. These terms can drive a phase transition in the system which can be studied using Monte Carlo methods 7,13 , perturbative expansions 14 or mappings to exactly solvable models 15 .…”

“…For free fermion systems this has led to an elegant 'periodic table'-like classification 3,4 , containing phases of matter such as the quantum spin Hall state 5,6 , the integer quantum Hall state and the p + ip superconductor. More generally the quest for observables that identify the type of topological order has spawned among others such quantities as topological entanglement entropies and spectra, whereas we have in previous work 7 proposed the topological S-matrix.…”

“…From a field theory point of view two families of models have been identified: Chern-Simons theories, which are closely related to the mathematics of knot theory and the Jones polynomial 9 , and discrete gauge theories 10 , of which Kitaev's toric code 8 is a close relative. Different lattice realizations of (non-chiral varieties of) these theories have been constructed, for example Levin-Wen models 11,12 or discrete lattice gauge theories 7 .…”