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 $G$ 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 $G$ , 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 ${\mathbb{Z}}_{2}$ symmetry which exchanges the $e$ and $m$ type anyons. We further illustrate our construction with a number of additional examples.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 $G$ 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 $G$, 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 ${\mathbb{Z}}_{2}$ symmetry which exchanges the $e$ and $m$ type anyons. We further illustrate our construction with a number of additional examples.

arXiv:1606.07816Physical review. B./Physical review. BDec 15, 2016127 citations

The String-Net Surface Code: Quantum Circuits for Doubled Topological Phases

Brendan Pankovich

Mar 6, 20191 citation

Topological Phases and Exactly Soluable Lattice Models

Chien-Hung Lin

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.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.

Dec 1, 20150 citations

Topics in Quantum-Mechanics with Applications to String Models

Humberto José

Jan 1, 19860 citations

G

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

G

, 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

{\mathbb{Z}}_{2}

symmetry which exchanges the

e

and

m

type anyons. We further illustrate our construction with a number of additional examples.