Dual Dirac Liquid on the Surface of the Electron Topological Insulator

We discuss a non-Fermi liquid gapless metallic surface state of the topological band insulator. It has an odd number of gapless Dirac fermions coupled to a noncompact $U(1)$ gauge field. This can be viewed as a vortex dual to the conventional Dirac fermion surface state. This surface duality is a reflection of a bulk dual description discussed recently for the gauged topological insulator. All the other known surface states can be conveniently accessed from the dual Dirac liquid, including the surface quantum Hall state, the Fu-Kane superconductor, the gapped symmetric topological order and the “composite Dirac liquid.” We also discuss the physical properties of the dual Dirac liquid and its connection to the half-filled Landau level.

Popular Summary

There is a curious duality between superconductors and insulators in two dimensions. A superconductor can be described by fluctuating charge modes coupled with vortices that are gapped and hence form an insulator. If the vortices condense and form a superconductor, the charge fluctuation becomes gapped because of the Higgs mechanism, and the entire system becomes an insulator. We theoretically show here that this duality experiences an interesting twist on the surface of a Fu-Kane-Mele topological insulator.

The most well-known surface state of a topological insulator is the Dirac metal phase in which the electrons form a Dirac cone. In this study, we show that the surface also admits a vortex metal phase in which certain vortices become Dirac fermions and are coupled with fluctuating charge modes. The superconductor-insulator duality takes on a particularly illuminating form in this picture: When the vortex fermions form a simple superconductor, the entire surface becomes an insulator with non-Abelian topological order; when the vortex fermions form an insulator with non-Abelian topological order, the entire surface becomes a simple superconductor, but with some vortices associated with non-Abelian statistics. This one surface state—the so-called dual Dirac liquid that acts like a liquid metallic state—can access all other surface states. These findings help to elucidate recent results presented by Son in 2015 about a particle-hole symmetric theory of the composite Fermi liquid.

We expect that our findings will motivate additional studies of topological insulators.