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Physical review. B./Physical review. B

Anyon condensation in string-net models

Chien‐Hung Lin, F. J. Burnell

Published Sep 13, 2024• 6 citations
Publisher

Abstract

We study the condensation of Abelian bosons in string-net models by constructing a family of Hamiltonians that can be tuned through any such transition. We show that these Hamiltonians admit two exactly solvable, string-net limits: one deep in the uncondensed phase, described by an initial, uncondensed string-net Hamiltonian, and one deep in the condensed phase, described by a final, condensed string-net model. We give a systematic description of the condensed string-net model in terms of the uncondensed string-net and the data associated with the condensing Abelian bosons. Specifically, if the uncondensed string-net is described by a fusion category $\mathcal{C}$, we show how the string labels and fusion data of the fusion category $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathcal{C}}$ describing the condensed string-net can be obtained from that of $\mathcal{C}$ and the data describing the string operators that create the condensing boson. This construction generalizes previous approaches to anyon condensation in string-nets by allowing the condensation of arbitrary Abelian bosons, including chiral bosons in string-nets constructed from, for example, Chern-Simons theories, which describe time-reversal invariant bilayer states. This gives a method for obtaining the full data for string-nets without explicit time-reversal symmetry from such bilayer models. We illustrate our approach with several examples.

DOI
10.1103/physrevb.110.115127

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We study the condensation of Abelian bosons in string-net models by constructing a family of Hamiltonians that can be tuned through any such transition. We show that these Hamiltonians admit two exactly solvable, string-net limits: one deep in the uncondensed phase, described by an initial, uncondensed string-net Hamiltonian, and one deep in the condensed phase, described by a final, condensed string-net model. We give a systematic description of the condensed string-net model in terms of the uncondensed string-net and the data associated with the condensing Abelian bosons. Specifically, if the uncondensed string-net is described by a fusion category C\mathcal{C}C, we show how the string labels and fusion data of the fusion category C\ifmmode~\else˜\fi\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathcal{C}}C\ifmmode~\else˜\fi​ describing the condensed string-net can be obtained from that of C\mathcal{C}C and the data describing the string operators that create the condensing boson. This construction generalizes previous approaches to anyon condensation in string-nets by allowing the condensation of arbitrary Abelian bosons, including chiral bosons in string-nets constructed from, for example, Chern-Simons theories, which describe time-reversal invariant bilayer states. This gives a method for obtaining the full data for string-nets without explicit time-reversal symmetry from such bilayer models. We illustrate our approach with several examples.
We study condensation of abelian bosons in string-net models, by constructing a family of Hamiltonians that can be tuned through any such transition. We show that these Hamiltonians admit two exactly solvable, string-net limits: one deep in the uncondensed phase, described by an initial, uncondensed string net Hamiltonian, and one deep in the condensed phase, described by a final, condensed string net model. We give a systematic description of the condensed string net model in terms of the uncondensed string net and the data associated with the condensing abelian bosons. Specifically, if the uncondensed string net is described by a fusion category C\mathcal{C}C, we show how the string labels and fusion data of the fusion category C~\mathcal{\tilde{C}}C~ describing the condensed string net can be obtained from that of C\mathcal{C}C and the data describing the string oeprators that create the condensing boson. This construction generalizes previous approaches to anyon condensation in string nets, by allowing the condensation of arbitrary abelian bosons, including chiral bosons in string nets constructed from (for example) Chern-Simons theories, which describe time-reversal invariant bilayer states. This gives a method for obtaining the full data for string nets without explicit time-reversal symmetry from such bilayer models. We illustrate our approach with several examples.
We describe how to construct generalized string-net models, a class of exactly solvable lattice models that realize a large family of two-dimensional topologically ordered phases of matter. The ground states of these models can be thought of as superpositions of different ``string-net configurations,'' where each string-net configuration is a trivalent graph with labeled edges, drawn in the xyxyxy plane. What makes this construction more general than the original string-net construction is that, unlike the original construction, tetrahedral reflection symmetry is not assumed, nor is it assumed that the ground-state wave function $\mathrm{\ensuremath{\Phi}}$ is ``isotropic'': i.e., in the generalized setup, two string-net configurations X1,X2{X}_{1},{X}_{2}X1​,X2​ that can be continuously deformed into one another can have different ground-state amplitudes $\mathrm{\ensuremath{\Phi}}({X}_{1})\ensuremath{\ne}\mathrm{\ensuremath{\Phi}}({X}_{2})$. As a result, generalized string-net models can realize topological phases that are inaccessible to the original construction. In this paper, we provide a more detailed discussion of ground-state wave functions, Hamiltonians, and minimal self-consistency conditions for generalized string-net models than what exists in the previous literature. We also show how to construct string operators that create anyon excitations in these models, and we show how to compute the braiding statistics of these excitations. Finally, we derive necessary and sufficient conditions for generalized string-net models to have isotropic ground-state wave functions on the plane or the sphere, a property that may be useful in some applications.
A bstract We develop a comprehensive framework for realizing anyon condensation of topological orders within the string-net model by constructing a Hamiltonian that bridges the parent string-net model before and the child string-net model after anyon condensation. Our approach classifies all possible types of bosonic anyon condensation in any parent string-net model and identifies the basic degrees of freedom in the corresponding child models. The Kogut-Susskind lattice gauge theory model is a special case of our model if the full degrees of freedom of the model are truncated from SU(2) representations to quantum group SU(2) k . Compared with the traditional UMTC perspective of topological orders, our method offers a finer categorical description of anyon condensation at the microscopic level. We also explicitly represent relevant UMTC categorical entities characterizing anyon condensation through our model-based physical quantities, providing practical algorithms for calculating these categorical data.
Anyon Condensation In Symmetry-Enriched Topological Phases: GGG-Grading of Multifusion Categories
Although anyon condensation is a standard mechanism for relating topological orders, anyon condensation in symmetry-enriched topological (SET) phases is more intricate because the condensate must also be compatible with the global symmetry. We study symmetry-preserving anyon condensation in SET phases described by the enlarged Hu-Geer-Wu (HGW) string-net model with multifusion-category input data. We show that a GGG-preserving condensation is characterized by a compatible grading of the input multifusion category, and that this grading constructs the multifusion-category input of the child SET phase. To make this construction concrete, we consider the case where the relevant data come from a finite group extension EEE of GGG by NNN: an EEE-graded fusion category induces a GGG-SET input by passing to the quotient symmetry GGG, and the resulting input naturally carries a compatible NNN-grading that implements the further condensation inside the SET phase while preserving GGG. We illustrate the construction using three quantum-double examples: the trivial extension Z2×Z2\mathbb{Z}_2\times\mathbb{Z}_2Z2​×Z2​, the non-Abelian semidirect product S3S_3S3​, and the nontrivial central extension Z4\mathbb{Z}_4Z4​. The Z4\mathbb{Z}_4Z4​ example further shows that symmetry fractionalized anyons do not obstruct symmetry-preserving condensation once the condensate is treated as a physical coherent state.
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)\text{U}(1)U(1) Chern-Simons theories, and we derive the KKK-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 $\ensuremath{\gamma},\ensuremath{\alpha}$ into the construction which are related to Z2{\mathbb{Z}}_{2}Z2​ and Z3{\mathbb{Z}}_{3}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.
We consider the string-net model on the honeycomb lattice for Ising anyons in the presence of a string tension. This competing term induces a nontrivial dynamics of the non-Abelian anyonic quasiparticles and may lead to a breakdown of the topological phase. Using high-order series expansions and exact diagonalizations, we determine the robustness of this doubled Ising phase which is found to be separated from two gapped phases. An effective quantum dimer model emerges in the large tension limit giving rise to two different translation symmetry-broken phases. Consequently, we obtain four transition points, two of which are associated to first-order transitions whereas the two others are found to be continuous and provide examples of recently proposed Bose condensation for anyons.