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String-net models in condensed matter systems

Curt W. von Keyserlingk

Published Jan 1, 2014• 0 citations
Publisher

Abstract

We study a class of three dimensional exactly solvable models of topological matter first put forward by Walker and Wang. While these are not models of interacting bosons or fermions, they may well capture the topological behaviour of some strongly correlated systems. In this work we give a full pedagogical treatment of a special simple case of these models, which we call the 3D semion model: We calculate its ground state degeneracies for a variety of boundary conditions, and classify its low-lying excitations. While point defects in the bulk are confined in pairs connected by energetic strings, the surface excitations are more interesting: the model has deconfined point defects pinned to the boundary of the lattice, and these exhibit semionic braiding statistics. The surface physics is reminiscent of a υ=1/2 bosonic fractional quantum Hall effect in its topological limit. Our special example of the 3D semion model captures much of the behaviour of more general `confined Walker-Wang models'. We contrast the 3D semion model with the closely related 3D version of the toric code (a lattice gauge theory) which has deconfined point excitations in the bulk. We discuss how more general Walker-Wang models can have a combination of fermionic or bosonic deconfined excitations in the bulk, and excitations which are deconfined on the surface but confined in the bulk. Having classified the spectra and surface properties of a number of these models, we construct a topological field theory which describes a subset of them exactly. Moreover, we argue that some Walker-Wang models capture the low-energy long-wavelength behaviour of certain gapped phases of axion electrodynamics which, in turn, are believed to be germane to certain condensed matter systems with strong spin-orbit coupling.

DOI
10.5287/ora-o82ompna4

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We study a class of three dimensional exactly solvable models of topological matter first put forward by Walker and Wang. While these are not models of interacting bosons or fermions, they may well capture the topological behaviour of some strongly correlated systems. In this work we give a full pedagogical treatment of a special simple case of these models, which we call the 3D semion model: We calculate its ground state degeneracies for a variety of boundary conditions, and classify its low-lying excitations. While point defects in the bulk are confined in pairs connected by energetic strings, the surface excitations are more interesting: the model has deconfined point defects pinned to the boundary of the lattice, and these exhibit semionic braiding statistics. The surface physics is reminiscent of a υ=1/2 bosonic fractional quantum Hall effect in its topological limit. Our special example of the 3D semion model captures much of the behaviour of more general `confined Walker-Wang models'. We contrast the 3D semion model with the closely related 3D version of the toric code (a lattice gauge theory) which has deconfined point excitations in the bulk. We discuss how more general Walker-Wang models can have a combination of fermionic or bosonic deconfined excitations in the bulk, and excitations which are deconfined on the surface but confined in the bulk. Having classified the spectra and surface properties of a number of these models, we construct a topological field theory which describes a subset of them exactly. Moreover, we argue that some Walker-Wang models capture the low-energy long-wavelength behaviour of certain gapped phases of axion electrodynamics which, in turn, are believed to be germane to certain condensed matter systems with strong spin-orbit coupling.
We study a class of three-dimensional (3D) exactly solvable models of topological matter first put forward by Walker and Wang [Front. Phys. 7(2) 150 (2012)]. While these are not models of interacting fermions, they may well capture the topological behavior of some strongly correlated systems. In this work, we give a full pedagogical treatment of a special simple case of these models, which we call the 3D semion model: We calculate its ground-state degeneracies for a variety of boundary conditions, and classify its low-lying excitations. While point defects in the bulk are confined in pairs connected by energetic strings, the surface excitations are more interesting: the model has deconfined point defects pinned to the boundary of the lattice, and these exhibit semionic braiding statistics. The surface physics is reminiscent of a $\ensuremath{\nu}=\frac{1}{2}$ bosonic fractional quantum Hall effect in its topological limit, and these considerations help motivate an effective field theoretic description for the lattice models as variants of bFbFbF theories. Our special example of the 3D semion model captures much of the behavior of more general ``confined Walker-Wang models.'' We contrast the 3D semion model with the closely related 3D version of the toric code (a lattice gauge theory) which has deconfined point excitations in the bulk, and we discuss how more general models may have some confined and some deconfined excitations. Having seen that there exist lattice models whose surfaces have the same topological order as a bosonic fractional quantum Hall effect on a confining bulk, we construct a lattice model whose surface has similar topological order to a fermionic quantum Hall effect. We find that in these models a fermion is always deconfined in the three-dimensional bulk.
We study the phase diagrams of a family of three-dimensional Walker-Wang--type lattice models, which are not topologically ordered but have deconfined anyonic excitations confined to their surfaces. We add a perturbation (analogous to that which drives the confining transition in Zp{\mathbb{Z}}_{p}Zp​ lattice gauge theories) to the Walker-Wang Hamiltonians, driving a transition in which all or some of the variables associated with the loop-gas or string-net ground states of these models become confined. We show that in many cases the location and nature of the phase transitions involved is exactly that of a generalized Zp{\mathbb{Z}}_{p}Zp​ lattice gauge theory, and use this to deduce the basic structure of the phase diagram. We further show that the relationship between the phases on opposite sides of the transition is fundamentally different than in conventional gauge theories: in the Walker-Wang case, the number of species of excitations that are deconfined in the bulk can increase across a transition that confines only some of the species of loops or string nets. The analog of the confining transition in the Walker-Wang models can therefore lead to bulk deconfinement and topological order.
Three-dimensional gauge theories with a discrete gauge group can emerge from spin models as a gapped topological phase with fractional point excitations (gauge charge) and loop excitations (gauge flux). It is known that 3D gauge theories can be ``twisted,'' in the sense that the gauge flux loops can have nontrivial braiding statistics among themselves and such twisted gauge theories are realized in models discovered by Dijkgraaf and Witten. A different framework to systematically construct three-dimensional topological phases was proposed by Walker and Wang and a series of examples have been studied. Can the Walker-Wang construction be used to realize the topological order in twisted gauge theories? This is not immediately clear because the Walker-Wang construction is based on a loop condensation picture while the Dijkgraaf-Witten theory is based on a membrane condensation picture. In this paper, we show that the answer to this question is Yes, by presenting an explicit construction of the Walker-Wang models which realize both the twisted and untwisted gauge theories with gauge group ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$. We identify the topological order of the models by performing modular transformations on the ground-state wave functions and show that the modular matrices exactly match those for the ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ gauge theories. By relating the Walker-Wang construction to the Dijkgraaf-Witten construction, our result opens up a way to study twisted gauge theories with fermonic charges, and correspondingly strongly interacting fermionic symmetry protected topological phases and their surface states, through exactly solvable models.
Topological phases of matter are usually realized in deconfined phases of gauge theories. In this context, confined phases with strongly fluctuating gauge fields seem to be irrelevant to the physics of topological phases. For example, the low-energy theory of the two-dimensional (2D) toric code model (i.e., the deconfined phase of Z2{\mathbb{Z}}_{2}Z2​ gauge theory) is a $\text{U}(1)\ifmmode\times\else\texttimes\fi{}\text{U}(1)$ Chern-Simons theory in which gauge charges (i.e., eee and mmm particles) are deconfined and the gauge fields are gapped, while the confined phase is topologically trivial. In this paper, we point out a route to constructing exotic three-dimensional (3D) gapped fermionic phases in a confining phase of a gauge theory. Starting from a parton construction with strongly fluctuating compact $\text{U}(1)\ifmmode\times\else\texttimes\fi{}\text{U}(1)$ gauge fields, we construct gapped phases of interacting fermions by condensing two linearly independent bosonic composite particles consisting of partons and $\text{U}(1)\ifmmode\times\else\texttimes\fi{}\text{U}(1)$ magnetic monopoles. This can be regarded as a 3D generalization of the 2D Bais-Slingerland condensation mechanism. Charge fractionalization results from a Debye-H\"uckel--type screening cloud formed by the condensed composite particles. Within our general framework, we explore two aspects of symmetry-enriched 3D Abelian topological phases. First, we construct a new fermionic state of matter with time-reversal symmetry and $\mathrm{\ensuremath{\Theta}}\ensuremath{\ne}\ensuremath{\pi}$, the fractional topological insulator. Second, we generalize the notion of anyonic symmetry of 2D Abelian topological phases to the charge-loop excitation symmetry (Charles\mathsf{Charles}Charles) of 3D Abelian topological phases. We show that line twist defects, which realize Charles\mathsf{Charles}Charles transformations, exhibit non-Abelian fusion properties.
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 connect a family of gauge theories (Maxwell theories with a magnetoelectric coupling $\ensuremath{\theta}=2\ensuremath{\pi}k,k\ensuremath{\in}\mathbb{Z}$) to the family of 3D topological lattice models introduced by Walker and Wang. In particular, we show that the lattice Hamiltonians capture a certain strong-coupling limit of these gauge theories, in which the system enters a gapped (confined) phase. We discuss the relationship between the topological order exhibited by certain of these lattice Hamiltonians and the characteristic electromagnetic response of the symmetry-protected bosonic topological insulator.