Edge states for topological insulators in two dimensions and their Luttinger-like liquids

TL;DR

This paper classifies edge states of two-dimensional topological insulators, explores their symmetry-protected zero modes, and studies how interactions lead to various Luttinger liquid behaviors on the edges.

Contribution

It extends the classification of 1D edge Hamiltonians for 2D topological insulators and analyzes the effects of interactions on these edge states.

Findings

Identified 17 inequivalent classes of edge Hamiltonians with 11 having protected zero modes.

Conjectured the existence of new topological insulator classes based on additional edge states.

Showed that symmetry-preserving interactions are exactly marginal, leading to diverse Luttinger liquids.

Abstract

Topological insulators in three spatial dimensions are known to possess a precise bulk/boundary correspondence, in that there is a one-to-one correspondence between the 5 classes characterized by bulk topological invariants and Dirac hamiltonians on the boundary with symmetry protected zero modes. This holographic characterization of topological insulators is studied in two dimensions. Dirac hamiltonians on the one dimensional edge are classified according to the discrete symmetries of time-reversal, particle-hole, and chirality, extending a previous classification in two dimensions. We find 17 inequivalent classes, of which 11 have protected zero modes. Although bulk topological invariants are thus far known for only 5 of these classes, we conjecture that the additional 6 describe edge states of new classes of topological insulators. The effects of interactions in two dimensions are also studied. We show that all interactions that preserve the symmetries are exactly marginal, i.e. preserve the gaplessness. This leads to a description of the distinct variations of Luttinger liquids that can be realized on the edge.