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arXiv:1405.2988 · Physical Review B

Walker-Wang models and axion electrodynamics

Curt W. von Keyserlingk, F. J. Burnell

Published Jan 26, 2015• 19 citations
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Abstract

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.

DOI
10.1103/physrevb.91.045134

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We connect a family of gauge theories (Maxwell theories with a magnetoelectric coupling \ensuremathθ=2\ensuremathπk,k\ensuremath∈Z\ensuremath{\theta}=2\ensuremath{\pi}k,k\ensuremath{\in}\mathbb{Z}\ensuremathθ=2\ensuremathπk,k\ensuremath∈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.
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.
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.
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.
A long-standing problem in the study of topological phases of matter has been to understand the types of fractional topological insulator (FTI) phases possible in 3+1 dimensions. Unlike ordinary topological insulators of free fermions, FTI phases are characterized by fractional \Theta Θ -angles, long-range entanglement, and fractionalization. Starting from a simple family of \mathbb{Z}_N ℤ N lattice gauge theories due to Cardy and Rabinovici, we develop a class of FTI phases based on the physical mechanism of oblique confinement and the modern language of generalized global symmetries. We dub these phases oblique topological insulators. Oblique TIs arise when dyons—bound states of electric charges and monopoles—condense, leading to FTI phases characterized by topological order, emergent one-form symmetries, and gapped boundary states not realizable in 2+1-D alone. Based on the lattice gauge theory, we present continuum topological quantum field theories (TQFTs) for oblique TI phases involving fluctuating one-form and two-form gauge fields. We show explicitly that these TQFTs capture both the generalized global symmetries and topological orders seen in the lattice gauge theory. We also demonstrate that these theories exhibit a universal “generalized magnetoelectric effect” in the presence of two-form background gauge fields. Moreover, we characterize the possible boundary topological orders of oblique TIs, finding a new set of boundary states not studied previously for these kinds of TQFTs.
The interplay of symmetry, topology, and many-body effects in the classification of phases of matter poses a formidable challenge in condensed-matter physics. Such many-body effects are typically induced by inter-particle interactions involving an action at a distance, such as the Coulomb interaction between electrons in a symmetry-protected topological (SPT) phase. In this work, we show that similar phenomena also occur in certain relativistic theories with interactions mediated by gauge bosons, and constrained by gauge symmetry. In particular, we introduce a variant of the Schwinger model or quantum electrodynamics (QED) in 1+11+11+1 dimensions on an interval, which displays dynamical edge states localized on the boundary. We show that the system hosts SPT phases with a dynamical contribution to the vacuum $\ensuremath{\theta}$-angle from edge states, leading to a new type of topological QED in 1+11+11+1 dimensions. The resulting system displays an SPT phase which can be viewed as a correlated version of the Su-Schrieffer-Heeger topological insulator for polyacetylene due to nonzero gauge couplings. We use bosonization and density-matrix renormalization group techniques to reveal the detailed phase diagram, which can further be explored in experiments of ultra-cold atoms in optical lattices.