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

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 five 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 five of these classes, we conjecture that the additional six 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 symmetry 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.