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arXiv:1611.08288 · Physical review. B./Physical review. B

Multiflavor string-net models

Chien‐Hung Lin

Published May 8, 2017• 2 citations
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Abstract

We generalize the string-net construction to multiple flavors of strings, each of which is labeled by the elements of an Abelian group ${G}_{i}$. The same flavor of strings can branch, while different flavors of strings can cross one another and thus they form intersecting string nets. We systematically construct the exactly soluble lattice Hamiltonians and the ground-state wave functions for the intersecting string-net condensed phases. We analyze the braiding statistics of the low-energy quasiparticle excitations and find that our model can realize all the topological phases as the string-net model with group $G={\ensuremath{\prod}}_{i}{G}_{i}$. In this respect, our construction provides various ways of building lattice models which realize topological order $G$, corresponding to different partitions of $G$ and thus different flavors of string nets. In fact, our construction concretely demonstrates the K\"unneth formula by constructing various lattice models with the same topological order. As an example, we construct the $G={\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ string-net model which realizes a non-Abelian topological phase by properly intersecting three copies of toric codes.

DOI
10.1103/physrevb.95.195110

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We construct exactly solvable models for a wide class of symmetry-enriched topological (SET) phases. Our construction applies to two-dimensional (2D) bosonic SET phases with finite unitary on-site symmetry group GGG and we conjecture that our models realize every phase in this class that can be described by a commuting projector Hamiltonian. Our models are designed so that they have a special property: If we couple them to a dynamical lattice gauge field with gauge group GGG, the resulting gauge theories are equivalent to string-net models. This property is what allows us to analyze our models in generality. As an example, we present a model for a phase with the same anyon excitations as the toric code and with a Z2{\mathbb{Z}}_{2}Z2​ symmetry which exchanges the eee and mmm type anyons. We further illustrate our construction with a number of additional examples.
In the past few years, the discovery of topological insulators has resurrected the study of exotic phases of matter beyond the Landau symmetry breaking paradigm. In this thesis, we focus on two particular classes of exotic phases, known as "topological phases" and "symmetry protected topological phases (SPT phases)" in 2D and 3D. Topological phases are generalizations of fractional quantum Hall liquids while SPT phases are generalizations of topological insulators. We investigate topological phases and SPT phases by constructing and analyzing appropriate exactly soluble lattice spin models. We address the following three questions: The first question involves a large class of exactly soluble lattice spin models called string-net models. It is known that string-net models can realize a large class of 2D topological phases, but it is unclear what topological phases can and can not be realized by these models. Thus our first question is: what are the most general topological phases that can be realized by string-net models? We find that a topological phase can be realized by a string-net model if and only if it supports a gapped edge, i.e. the edge can be gapped by suitable local interactions. In the second question we ask: how can we distinguish different 3D topological phases? While it is well known that 3D topological phases can be distinguished by the braiding statistics between particle and loop-like excitations, we construct two exactly soluble lattice models that demonstrate that we also have to consider the braiding statistics of loops with other loops. In the final part of the thesis, we derive a bulk-boundary correspondence for 3D SPT phases. More specifically, we prove that a particular exactly soluble lattice spin model has protected surface states using the fact that the vortex loop excitations in the bulk have non-trivial braiding statistics. We prove the result for a particular 3D exactly soluble lattice model, but our arguments apply more generally.
We show that quantum systems of extended objects naturally give rise to a large class of exotic phases---namely topological phases. These phases occur when extended objects, called ``string-nets,'' become highly fluctuating and condense. We construct a large class of exactly soluble 2D spin Hamiltonians whose ground states are string-net condensed. Each ground state corresponds to a different parity invariant topological phase. The models reveal the mathematical framework underlying topological phases: tensor category theory. One of the Hamiltonians---a spin-1∕21∕21∕2 system on the honeycomb lattice---is a simple theoretical realization of a universal fault tolerant quantum computer. The higher dimensional case also yields an interesting result: we find that 3D string-net condensation naturally gives rise to both emergent gauge bosons and emergent fermions. Thus, string-net condensation provides a mechanism for unifying gauge bosons and fermions in 3 and higher dimensions.
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 $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathcal{C}}$ 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.