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

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.

Figure 1

Summary of some simple integer bosonic topological phases. (a) A chiral phase of bosons (no symmetry required). An integer multiple of eight chiral bosons at the edge is needed to evade topological order, leading to a quantized thermal Hall conductance ${\kappa }_{xy}/T=8n{L}_0$ in units of the the universal thermal conductance $L_0=\frac{{\pi }^2{k}_B^2}{3h}$. These are bosonic analogs of chiral superconductors. (b) A nonchiral phase of bosons protected by $U\left(1\right)$ symmetry (e.g., charge conservation). Distinct phases can be labeled by the quantized Hall conductance ${\sigma }_{xy}=2n{\sigma }_0$, which are even integer multiples of the universal conductance ${\sigma }_0=q^2/h$ for particles with charge $q$. These are bosonic analogs of the integer quantum Hall phases. (c) A nonchiral phase stabilized in the presence of time reversal and $U\left(1\right)$ charge conservation symmetries, the same symmetries used to define quantum spin Hall (topological) insulators. A $Z_2$ topological classification is obtained, although bosonic time reversal that squares to $+1$ is involved.

Figure 2

Schematic illustration of interwire coupling terms which stabilize the bosonic SPT phases protected by $U\left(1\right)$ symmetry, with Hall conductance ${\sigma }_{xy}=2q$. Solid horizontal lines stand for quantum wires of charged bosons [each carries unit $U\left(1\right)$ charge], while dashed horizontal lines represent quantum wires composed of neutral (say spin) degrees of freedom. Dashed and solid arrows illustrate the two interwire coupling terms in Eq. (130) that gap the bulk, but leave behind nontrivial edge states.