TY - JOUR

T1 - Generalizations and limitations of string-net models

AU - Lin, Chien Hung

AU - Levin, Michael

PY - 2014/5/21

Y1 - 2014/5/21

N2 - We ask which topological phases can and can not be realized by exactly soluble string-net models. We answer this question for the simplest class of topological phases, namely, those with Abelian braiding statistics. Specifically, we find that an Abelian topological phase can be realized by a string-net model if and only if (i) it has a vanishing thermal Hall conductance and (ii) it has at least one Lagrangian subgroup, a subset of quasiparticles with particular topological properties. Equivalently, we find that an Abelian topological phase is realizable if and only if it supports a gapped edge. We conjecture that the latter criterion generalizes to the non-Abelian case. We establish these results by systematically constructing all possible Abelian string-net models and analyzing the quasiparticle braiding statistics in each model. We show that the low-energy effective field theories for these models are multicomponent U(1) Chern-Simons theories, and we derive the K-matrix description of each model. An additional feature of this work is that the models we construct are more general than the original string-net models, due to several new ingredients. First, we introduce two new objects γ,α into the construction which are related to Z2 and Z3 Frobenius-Schur indicators. Second, we do not assume parity invariance. As a result, we can realize topological phases that were not accessible to the original construction, including phases that break time-reversal and parity symmetry.

AB - We ask which topological phases can and can not be realized by exactly soluble string-net models. We answer this question for the simplest class of topological phases, namely, those with Abelian braiding statistics. Specifically, we find that an Abelian topological phase can be realized by a string-net model if and only if (i) it has a vanishing thermal Hall conductance and (ii) it has at least one Lagrangian subgroup, a subset of quasiparticles with particular topological properties. Equivalently, we find that an Abelian topological phase is realizable if and only if it supports a gapped edge. We conjecture that the latter criterion generalizes to the non-Abelian case. We establish these results by systematically constructing all possible Abelian string-net models and analyzing the quasiparticle braiding statistics in each model. We show that the low-energy effective field theories for these models are multicomponent U(1) Chern-Simons theories, and we derive the K-matrix description of each model. An additional feature of this work is that the models we construct are more general than the original string-net models, due to several new ingredients. First, we introduce two new objects γ,α into the construction which are related to Z2 and Z3 Frobenius-Schur indicators. Second, we do not assume parity invariance. As a result, we can realize topological phases that were not accessible to the original construction, including phases that break time-reversal and parity symmetry.

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U2 - 10.1103/PhysRevB.89.195130

DO - 10.1103/PhysRevB.89.195130

M3 - Article

AN - SCOPUS:84901397983

VL - 89

JO - Physical Review B

JF - Physical Review B

SN - 2469-9950

IS - 19

M1 - 195130

ER -