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 $\text{U}(1)$ 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 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) 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 U(1)\ensuremath⋊Z2 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 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 Z2 symmetry) can exhibit a Z2-breaking quantum Hall effect on their boundaries, with this boundary quantum Hall effect described by a Chern-Simons term at level 2m!. 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 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 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) principal chiral model with a topological $\mathrm{\ensuremath{\Theta}}$ term, whose boundary is described by a Sp(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=1 is equivalent to the familiar O(4) nonlinear sigma model (NLSM) with a topological term, whose boundary is a (1+1)D conformal field theory with central charge c=1. After breaking the O(4) symmetry to its subgroups, this model can be viewed as bosonic SPT states with U(1), or 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 conformal field theory with gapless bosonic modes.
By definition, the physics of the d-dimensional (dim) boundary of a (d+1)-dim symmetry protected topological (SPT) state cannot be realized as itself on a d-dim lattice. If the symmetry of the system is unitary, then a formal way to determine whether a d-dim theory must be a boundary or not, is to couple this theory to a gauge field (or to ``gauge'' its symmetry), and check if there is a gauge anomaly. In this paper we discuss the following question: Can the boundary of a SPT state be driven into a fully gapped topological order which preserves all the symmetries? We argue (conjecture) that if the gauge anomaly of the boundary is ``perturbative,'' then the boundary must remain gapless; while if the boundary only has global gauge anomaly but no perturbative anomaly, then it is possible to gap out the boundary by driving it into a topological state, when $d\ensuremath{\ge}2$. We will demonstrate this conjecture with two examples: (1) the 3d spin-1/2 chiral fermion with the well-known Witten's global anomaly [Phys. Lett. 117, 324 (1982)], which can be realized on the boundary of a 4d topological superconductor with SU(2) or $U(1)\ensuremath{\rtimes}{Z}_{2}$ symmetry; and (2) the 4d boundary of a 5d topological superconductor with the same symmetry. We show that these boundary systems can be driven into a fully gapped Z2N topological order with topological degeneracy, but this Z2N topological order cannot be future driven into a trivial confined phase that preserves all the symmetries due to some special properties of its topological defects. Our study also leads to exotic states of matter in pure 3d space.
We present the correspondence between symmetry-protected topological (SPT) phases and their anomalous boundary states, based on examples in various spacetime dimensions. Through the study of the effect of interactions on these SPT phases, we discovered a new formalism of quantum anomalies, associated with discrete spacetime (such as time-reversal and spatial reflection) symmetries in particular, to classify distinct interacting topological phases. An example is the Z2 classification of the (2+1)d topological insulator protected by charge U(1) and time-reversal (or CP) symmetries, which can be deduced by the form of the global U(1) gauge anomalies on its edge theories defined on closed unorientable manifolds. In this case, the nontrivial phase (in free systems) is robust against electron interactions. Another example is the (3+1)d topological superconductor protected by only time-reversal or reflection symmetry. For this system, we identified the bulk phase by studying the global gravitational anomalies of the surface theories formulated on unorientable spacetime manifolds, and also discussed its connection to the collapse of the non-interacting classification by an integer Z to Z16, in the presence of interactions. We also revisit the problem of gauging a discrete internal symmetry in theories of chiral (Weyl) fermions in 3+1 dimensions – which have not been fully understood so far – from the perspective of fermionic SPT phases in 4+1 dimensions. Comparing with the previous results, we give a complete answer for the anomalies constraints on the discrete symmetry, as our approach is based on purely geometrical considerations, namely, our assumption is more fundamental and general. Furthermore, our result also provides an understanding of gapped states of fermions with anomalous discrete symmetries, and we present a model, based on weak coupling, for constructing these anomalous gapped states.
Quantum anomalies arise when symmetries of a classical theory cannot be preserved upon quantization, leading to unconventional topological responses. A prominent example is the parity anomaly of a single two-dimensional Dirac fermion, which enforces a half-quantized Hall response. Anomaly inflow mechanism allows this effect to be observed at the surfaces of three-dimensional topological insulators, however, its realization in a genuinely two-dimensional system has remained elusive. Here we report the observation of a parity-anomalous Hall response at the critical point of a quantum Hall topological phase transition in a synthetic two-dimensional system of ultracold dysprosium atoms. By coupling a continuous spatial dimension to a finite synthetic dimension encoded in atomic spin states, we engineer tunable Chern bands with C = 0 and 1. At the transition, the bulk gap closes at a single Dirac point, where we observe a robust half-quantized Hall drift despite strong non-adiabatic excitations. We show that this response originates from the global structure of the band topology, is protected by an emergent parity symmetry at criticality, and disappears when parity is explicitly broken. Our work establishes synthetic quantum systems as a powerful platform to probe quantum anomalies and their interplay with topology and non-equilibrium dynamics.
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 K-matrix description of quantum Hall states to classify such ``integer'' topological phases. Our main result is a general formalism that incorporates symmetries into the K-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 K matrix is the Cartan matrix of Lie group 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.