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arXiv:1205.3156 · Physical Review B

Theory and classification of interacting integer topological phases in two dimensions: A Chern-Simons approach

Yuan-Ming Lu, Ashvin Vishwanath

Published Sep 14, 2012• 507 citations
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Abstract

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 ${E}_{8}$. 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.

DOI
10.1103/physrevb.86.125119

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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.
Integer quantum Hall states and their generalization to topological insulators and superconductors are paradigmatic examples of invertible fermionic topological states of matter. Here, the authors develop a comprehensive characterization and classification of invertible fermionic topological phases of matter in two spatial dimensions for general symmetry groups. The results are nonperturbative and apply to strongly interacting systems. In particular, they extend previous classification results to account for the possible chiral nature of invertible phases.
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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 $2m\ensuremath{-}1$ $(m=1,2,\ensuremath{\cdots})$, 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!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.
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