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arXiv:1611.03504 · Physical review. B./Physical review. B

Topological electromagnetic responses of bosonic quantum Hall, topological insulator, and chiral semimetal phases in all dimensions

Matthew F. Lapa, Chao‐Ming Jian, Peng Ye, Taylor L. Hughes

Published Jan 27, 2017• 29 citations
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Abstract

We calculate the topological part of the electromagnetic response of bosonic integer quantum Hall (BIQH) phases in odd (space-time) dimensions, and bosonic topological insulator (BTI) and bosonic chiral semimetal (BCSM) phases in even dimensions. To do this, we use the nonlinear sigma model (NLSM) description of bosonic symmetry-protected topological (SPT) phases, and the method of gauged Wess-Zumino (WZ) actions. We find the surprising result that for BIQH states in dimension $2m\ensuremath{-}1$ $(m=1,2,\ensuremath{\cdots})$, the bulk response to an electromagnetic field ${A}_{\ensuremath{\mu}}$ is characterized by a Chern-Simons term for ${A}_{\ensuremath{\mu}}$ with a level quantized in integer multiples of $m!$ (factorial). We also show that BTI states (which have an extra ${\mathbb{Z}}_{2}$ symmetry) can exhibit a ${\mathbb{Z}}_{2}$-breaking quantum Hall effect on their boundaries, with this boundary quantum Hall effect described by a Chern-Simons term at level $\frac{m!}{2}$. We show that the factor of $m!$ can be understood by requiring gauge invariance of the exponential of the Chern-Simons term on a general Euclidean manifold, and we also use this argument to characterize the electromagnetic and gravitational responses of fermionic SPT phases with $\text{U}(1)$ symmetry in all odd dimensions. We then use our gauged boundary actions for the BIQH and BTI states to (i) construct a bosonic analog of a chiral semimetal (BCSM) in even dimensions, (ii) show that the boundary of the BTI state exhibits a bosonic analog of the parity anomaly of Dirac fermions in odd dimensions, and (iii) study anomaly inflow at domain walls on the boundary of BTI states. In a series of Appendixes we derive important formulas and additional results. In particular, in Appendix A we use the connection between equivariant cohomology and gauged WZ actions to give a mathematical interpretation of the actions for the BIQH and BTI boundaries constructed in this paper.

DOI
10.1103/physrevb.95.035149

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We calculate the topological part of the electromagnetic response of bosonic integer quantum Hall (BIQH) phases in odd (space-time) dimensions, and bosonic topological insulator (BTI) and bosonic chiral semimetal (BCSM) phases in even dimensions. To do this, we use the nonlinear sigma model (NLSM) description of bosonic symmetry-protected topological (SPT) phases, and the method of gauged Wess-Zumino (WZ) actions. We find the surprising result that for BIQH states in dimension 2m\ensuremath−12m\ensuremath{-}12m\ensuremath−1 (m=1,2,\ensuremath⋯)(m=1,2,\ensuremath{\cdots})(m=1,2,\ensuremath⋯), the bulk response to an electromagnetic field A\ensuremathμ{A}_{\ensuremath{\mu}}A\ensuremathμ​ is characterized by a Chern-Simons term for A\ensuremathμ{A}_{\ensuremath{\mu}}A\ensuremathμ​ with a level quantized in integer multiples of m!m!m! (factorial). We also show that BTI states (which have an extra Z2{\mathbb{Z}}_{2}Z2​ symmetry) can exhibit a Z2{\mathbb{Z}}_{2}Z2​-breaking quantum Hall effect on their boundaries, with this boundary quantum Hall effect described by a Chern-Simons term at level m!2\frac{m!}{2}2m!​. We show that the factor of m!m!m! can be understood by requiring gauge invariance of the exponential of the Chern-Simons term on a general Euclidean manifold, and we also use this argument to characterize the electromagnetic and gravitational responses of fermionic SPT phases with U(1)\text{U}(1)U(1) symmetry in all odd dimensions. We then use our gauged boundary actions for the BIQH and BTI states to (i) construct a bosonic analog of a chiral semimetal (BCSM) in even dimensions, (ii) show that the boundary of the BTI state exhibits a bosonic analog of the parity anomaly of Dirac fermions in odd dimensions, and (iii) study anomaly inflow at domain walls on the boundary of BTI states. In a series of Appendixes we derive important formulas and additional results. In particular, in Appendix A we use the connection between equivariant cohomology and gauged WZ actions to give a mathematical interpretation of the actions for the BIQH and BTI boundaries constructed in this paper.
We study perturbative and global anomalies at the boundaries of bosonic analogs of integer quantum Hall (BIQH) and topological insulator (BTI) phases using a description of the boundaries of these phases in terms of a nonlinear sigma model (NLSM) with Wess-Zumino term. One of the main results of the paper is that these anomalies are robust against arbitrary smooth deformations of the target space of the NLSM which describes the phase, provided that the deformations also respect the symmetry of the phase. In the first part of the paper, we discuss the perturbative U(1)\text{U}(1)U(1) anomaly at the boundary of BIQH states in all odd (space-time) dimensions. In the second part, we study global anomalies at the boundary of BTI states in even dimensions. In a previous work [Lapa et al., Phys. Rev. B 95, 035149 (2017)] we argued that the boundary of the BTI phase exhibits a global anomaly which is an analog of the parity anomaly of Dirac fermions in three dimensions. Here, we elevate this argument to a proof for the boundary of the two-dimensional BTI state by explicitly computing the partition function of the gauged NLSM describing the boundary. We then use the powerful equivariant localization technique to show that this global anomaly is robust against all smooth deformations of the target space of the NLSM which preserve the $\text{U}(1)\ensuremath{\rtimes}{\mathbb{Z}}_{2}$ symmetry of the BTI state. We also comment on the difficulties of generalizing this latter proof to higher dimensions. Finally, we discuss the expected low-energy behavior of the boundary theories studied in this paper when the coupling constants are allowed to flow under the renormalization group.
We study topological phases of interacting systems in two spatial dimensions in the absence of topological order (i.e., with a unique ground state on closed manifolds and no fractional excitations). These are the closest interacting analogs of integer quantum Hall states, topological insulators, and superconductors. We adapt the well-known Chern-Simons KKK-matrix description of quantum Hall states to classify such ``integer'' topological phases. Our main result is a general formalism that incorporates symmetries into the KKK-matrix description. Remarkably, this simple analysis yields the same list of topological phases as a recent group cohomology classification, and in addition provides field theories and explicit edge theories for all these phases. The bosonic topological phases, which only appear in the presence of interactions and which remain well defined in the presence of disorder, include (i) bosonic insulators with a Hall conductance quantized to even integers, (ii) a bosonic analog of quantum spin Hall insulators, and (iii) a bosonic analog of a chiral topological superconductor, whose KKK matrix is the Cartan matrix of Lie group E8{E}_{8}E8​. We also discuss interacting fermion systems where symmetries are realized in a projective fashion, where we find the present formalism can handle a wider range of symmetries than a recent group super-cohomology classification. Lastly, we construct microscopic models of these phases from coupled one-dimensional systems.
We construct a bosonic analog of a two-dimensional topological Dirac semimetal (DSM). The low-energy description of the most basic 2D DSM model consists of two Dirac cones at positions $\ifmmode\pm\else\textpm\fi{}{\mathbf{k}}_{0}$ in momentum space. The local stability of the Dirac cones is guaranteed by a composite symmetry Z2TI{Z}_{2}^{\mathcal{TI}}Z2TI​, where T\mathcal{T}T is time reversal and I\mathcal{I}I is inversion. This model also exhibits interesting time-reversal and inversion symmetry breaking electromagnetic responses. In this work we construct a bosonic version by replacing each Dirac cone with a copy of the O(4)O(4)O(4) nonlinear sigma model (NLSM) with topological theta term and theta angle $\ensuremath{\theta}=\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}$. One copy of this NLSM also describes the gapless surface termination of the 3D bosonic topological insulator (BTI). We compute the time-reversal and inversion symmetry breaking electromagnetic responses for our model and show that they are twice the value one gets in the DSM case matching what one might expect from, for example, a bosonic Chern insulator. We also investigate the stability of the BSM model and find that the composite Z2TI{Z}_{2}^{\mathcal{TI}}Z2TI​ symmetry again plays an important role. Along the way we clarify many aspects of the surface theory of the BTI including the electromagnetic response, the charges and statistics of vortex excitations, and the stability to symmetry-allowed perturbations. We briefly comment on the relation between the various descriptions of the O(4)O(4)O(4) NLSM with $\ensuremath{\theta}=\ensuremath{\pi}$ used in this paper (a dual vortex description and a description in terms of four massless fermions) and the recently proposed dual description of the BTI surface in terms of 2+12+12+1-dimensional quantum electrodynamics with two flavors of fermion (N=2QED3)(N=2 {\text{QED}}_{3})(N=2QED3​). In a set of four Appendixes we review some of the tools used in the paper and also derive some of the more technical results.
Symmetry-protected topological phases (SPT) are short-range entangled gapped states protected by global symmetry. Nontrivial SPT phases cannot be adiabatically connected to the trivial disordered state (or atomic insulator) as long as certain global symmetry GGG is unbroken. At low energies, most of the two-dimensional SPTs with Abelian symmetry can be described by topological quantum field theory (TQFT) of the multicomponent Chern-Simons type. However, in contrast to the fractional quantum Hall effect where TQFT can give rise to interesting bulk anyons, TQFT for SPTs only supports trivial bulk excitations. The essential question in TQFT descriptions for SPTs is to understand how the global symmetry is implemented in the partition function. In this paper, we systematically study TQFT of three-dimensional SPTs with unitary Abelian symmetry (e.g., ${\mathbb{Z}}_{{N}_{1}}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{{N}_{2}}\ifmmode\times\else\texttimes\fi{}...$). In addition to the usual multicomponent BFBFBF topological term at level-1, we find that there are new topological terms with quantized coefficients (e.g., ${a}^{1}\ensuremath{\wedge}{a}^{2}\ensuremath{\wedge}d{a}^{2}$ and ${a}^{1}\ensuremath{\wedge}{a}^{2}\ensuremath{\wedge}{a}^{3}\ensuremath{\wedge}{a}^{4}$) in TQFT actions, where a1,a2,...{a}^{1},{a}^{2},...a1,a2,... are 1-form U(1) gauge fields. These additional topological terms cannot be adiabatically turned off as long as GGG is unbroken. By investigating symmetry transformations for the TQFT partition function, we end up with the classification of SPTs that is consistent with the well-known group cohomology approach. We also discuss how to gauge the global symmetry and possible TQFT descriptions of Dijkgraaf-Witten gauge theory.
We analyze 2+1d2+1d2+1d and 3+1d3+1d3+1d bosonic symmetry protected topological (SPT) phases of matter protected by onsite symmetry group GGG by using dual bulk and boundary approaches. In the bulk, we study an effective field theory, which upon coupling to a background flat GGG gauge field furnishes a purely topological response theory. The response action evaluated on certain manifolds, with appropriate choice of background gauge field, defines a set of SPT topological invariants. Further, SPTs can be gauged by summing over all isomorphism classes of flat GGG gauge fields to obtain Dijkgraaf-Witten topological GGG gauge theories. These topological gauge theories can be ungauged by first introducing and then proliferating defects that spoil the gauge symmetry. This mechanism is related to anyon condensation in 2+1d2+1d2+1d and condensing bosonic gauge charges in 3+1d3+1d3+1d. In the dual boundary approach, we study 1+1d1+1d1+1d and 2+1d2+1d2+1d quantum field theories that have GGG 't-Hooft anomalies that can be precisely canceled by (the response theory of) the corresponding bulk SPT. We show how to construct/compute topological invariants for the bulk SPTs directly from the boundary theories. Further, we sum over boundary partition functions with different background gauge fields to construct GGG characters that generate topological data for the bulk topological gauge theory. Finally, we study a 2+1d2+1d2+1d quantum field theory with a mixed ${\mathbb{Z}}_{2}^{T/R}\ifmmode\times\else\texttimes\fi{}\text{U}(1)$ anomaly where Z2T/R{\mathbb{Z}}_{2}^{T/R}Z2T/R​ is time-reversal/reflection symmetry, and the U(1)\text{U}(1)U(1) could be a 0-form or 1-form symmetry depending on the choice of time reversal/reflection action. We briefly discuss the bulk effective action and topological response for a theory in 3+1d3+1d3+1d that cancels this anomaly. This signals the existence of SPTs in 3+1d3+1d3+1d protected by 0,1-form $\text{U}(1)\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}^{T,R}$.
We propose a series of simple two-dimensional (2D) lattice interacting fermion models that we demonstrate at low energy describe bosonic symmetry-protected topological (SPT) states and quantum phase transitions between them. This is because due to interaction, the fermions are gapped both at the boundary of the SPT states and at the bulk quantum phase transition, thus these models at low energy can be described completely by bosonic degrees of freedom. We show that the bulk of these models is described by a Sp(N\mathrm{Sp}(NSp(N) principal chiral model with a topological $\mathrm{\ensuremath{\Theta}}$ term, whose boundary is described by a Sp(N\mathrm{Sp}(NSp(N) principal chiral model with a Wess-Zumino-Witten term at level 1. The quantum phase transition between SPT states in the bulk is tuned by a particular interaction term, which corresponds to tuning $\mathrm{\ensuremath{\Theta}}$ in the field theory, and the phase transition occurs at $\mathrm{\ensuremath{\Theta}}=\ensuremath{\pi}$. The simplest version of these models with N=1N=1N=1 is equivalent to the familiar O(4) nonlinear sigma model (NLSM) with a topological term, whose boundary is a (1+1)D(1+1)\text{D}(1+1)D conformal field theory with central charge c=1c=1c=1. After breaking the O(4) symmetry to its subgroups, this model can be viewed as bosonic SPT states with U(1), or Z2{Z}_{2}Z2​ symmetries, etc. All of these fermion models, including the bulk quantum phase transitions, can be simulated with the determinant quantum Monte Carlo method without the sign problem. Recent numerical results strongly suggest that the quantum disordered phase of the O(4) NLSM with precisely $\mathrm{\ensuremath{\Theta}}=\ensuremath{\pi}$ is a stable (2+1)D(2+1)\text{D}(2+1)D conformal field theory with gapless bosonic modes.