Gapped symmetry preserving surface state for the electron topological insulator

It is well known that the three-dimensional (3D) electronic topological insulator (TI) with charge-conservation and time-reversal symmetry cannot have a trivial insulating surface that preserves symmetry. It is often implicitly assumed that if the TI surface preserves both symmetries then it must be gapless. Here we show that it is possible for the TI surface to be both gapped and symmetry preserving, at the expense of having surface-topological order. In contrast to analogous bosonic topological insulators, this symmetric surface topological order is intrinsically non-Abelian. We show that the surface-topological order provides a complete nonperturbative definition of the electron TI that transcends a free-particle band-structure picture, and could provide a useful perspective for studying strongly correlated topological Mott insulators.

Figure 1

Exchanging two $\frac{hc}{e}$ vortices at the superconducting surface of a TI slab (top panel) leads to a linking of their magnetic field lines, which gives a phase of $-1$, demonstrating that $\frac{hc}{e}$ vortices are semionic.

Figure 2

The nonfractionalized TR-breaking quantum Hall insulator (QHI) with coating the TI surface with a 2D TR-breaking topologically ordered state with ${\sigma }_H={\kappa }_H=\pm \frac{1}{2}$ (depicted in orange and purple, respectively), as explained in the text. The half-integer quantum Hall conductance can be seen by considering a domain between these two coatings as shown in the above figure for a spherical TI.