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

TL;DR

This paper develops a Chern-Simons K-matrix formalism to classify and describe interacting two-dimensional topological phases without fractional excitations, including bosonic and fermionic systems, extending previous classifications and providing explicit models.

Contribution

It introduces a symmetry-incorporated K-matrix classification for interacting topological phases, matching recent group cohomology results and offering explicit field and edge theories.

Findings

Classified bosonic topological phases with quantized Hall conductance

Identified bosonic analogs of quantum spin Hall and chiral topological superconductors

Constructed microscopic models from coupled one-dimensional systems

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.