Symmetry-respecting topologically ordered surface phase of three-dimensional electron topological insulators

A three-dimensional electron topological insulator (ETI) is a phase of matter protected by particle-number conservation and time-reversal symmetry. It was previously believed that the surface of an ETI must be gapless unless one of these symmetries is broken. A well-known symmetry-preserving, gapless surface termination of an ETI supports an odd number of Dirac cones. In this paper, we deduce a symmetry-respecting, gapped surface termination of an ETI, which carries an intrinsic two-dimensional (2d) topological order, Moore-Read x U(1)(-2). The Moore-Read sector supports non-Abelian charge 1/4 anyons, while the Abelian, U(1)(-2), (antisemion) sector is electrically neutral. Time-reversal symmetry is implemented in this surface phase in a highly nontrivial way. Moreover, it is impossible to realize this phase strictly in 2d, simultaneously preserving its implementation of both the particle-number and time-reversal symmetries. A one-dimensional (1d) edge on the ETI surface between the topologically ordered phase and the topologically trivial time-reversal-broken phase with a Hall conductivity sigma(xy) = 1/2 carries a right-moving neutral Majorana mode, a right-moving bosonic charge mode, and a left-moving bosonic neutral mode. The topologically ordered phase is separated from the surface superconductor by a direct second-order phase transition in the XY* universality class, which is driven by the condensation of a charge 1/2 boson, when approached from the topologically ordered side, and proliferation of a flux 4 pi (2hc/e) vortex, when approached from the superconducting side. In addition, we prove that time-reversal invariant (interacting) electron insulators with no intrinsic topological order and electromagnetic response characterized by a theta angle, theta = pi, do not exist if the electrons transform as Kramers singlets under time reversal.