Abstract A 3D fermionic topological insulator has a gapless Dirac surface state protected by time-reversal symmetry and charge conservation symmetry. The surface state can be gapped by introducing ferromagnetism to break time-reversal symmetry, introducing superconductivity to break charge conservation, or entering a topological phase. In this paper, we construct a minimal gapped topological phase that preserves both time-reversal and charge conservation symmetries and supports Ising-type non-Abelian anyons. This phase can be understood heuristically as emerging from a surface s-wave superconducting state via the condensation of eight-vortex composites. The topological phase inherits vortices supporting Majorana zero modes from the surface superconducting state. However, since it is time-reversal invariant, the surface topological phase is a distinct phase from the Ising topological phase, which can be viewed as a quantum-disordered spin-polarized p x + ip y superconductor. We discuss the anyon model of this topological phase and the manner in which time-reversal symmetry is realized in it. We also study the interfaces between the topological state and other surface gapped phases.
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, $\text{Moore-Read}\ifmmode\times\else\texttimes\fi{}\mathrm{U}{(1)}_{\ensuremath{-}2}$. The Moore-Read sector supports non-Abelian charge
1/4 anyons, while the Abelian, $\mathrm{U}{(1)}_{\ensuremath{-}2}$, (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 ${\ensuremath{\sigma}}_{xy}=1/2$ 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 $4\ensuremath{\pi}\phantom{\rule{0.28em}{0ex}}(2hc/e)$ 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 $\ensuremath{\theta}$ angle, $\ensuremath{\theta}=\ensuremath{\pi}$, do not exist if the electrons transform as Kramers singlets under time reversal.
Three dimensional topological superconductors with time reversal symmetry (class DIII) are indexed by an integer
ν, the number of surface Majorana cones, according to the free fermion classification. The superfluid B phase of He
3 realizes the
ν=1 phase. Recently, it has been argued that this classification is reduced in the presence of interactions to Z
16. This was argued from the symmetry respecting surface topological orders of these states, which provide a non-perturbative definition of the bulk topological phase. Here, we verify this conclusion by focusing on the even index case,
ν=2m, where a vortex condensation approach can be used to explicitly derive the surface topological orders. We show a direct relation to the well known result on one dimensional topological superconductors (class BDI), where interactions reduce the free fermion classification from Z down to Z
8. Finally, we discuss in detail the fermionic analog of Kramers time reversal symmetry, which allows semions of some surface topological orders to transform as
T2=±i.
The surfaces of three-dimensional topological insulators (3D TIs) are generally described as Dirac metals, with a single Dirac cone. It was previously believed that a gapped surface implied breaking of either time-reversal
T or
U(1) charge conservation symmetry. Here, we discuss a possibility in the presence of interactions, a surface phase that preserves all symmetries but is nevertheless gapped and insulating. Then, the surface must develop topological order of a kind that can not be realized in a two-dimensional (2D) system with the same symmetries. We discuss candidate surface states, non-Abelian quantum Hall states which, when realized in 2D, have ${\ensuremath{\sigma}}_{xy}=\frac{1}{2}$ and hence break
T symmetry. However, by constructing an exactly soluble 3D lattice model, we show they can be realized as
T-symmetric surface states. The corresponding 3D phases are confined, and have $\ensuremath{\theta}=\ensuremath{\pi}$ magnetoelectric response. Two candidate states have the same 12-particle topological order, the (Read-Moore) Pfaffian state with the neutral sector reversed, which we term T-Pfaffian topological order, but differ in their
T transformation. Although we are unable to connect either of these states directly to the superconducting TI surface, we argue that one of them describes the 3D TI surface, while the other differs from it by a bosonic topological phase. We also discuss the 24-particle Pfaffian-antisemion topological order (which can be connected to the superconducting TI surface) and demonstrate that it can be realized as a
T-symmetric surface state.
We show that the chiral Dirac and Majorana hinge modes in three-dimensional higher-order topological insulators (HOTIs) and superconductors (HOTSCs) can be gapped while preserving the protecting C_{2n}T symmetry upon the introduction of non-Abelian surface topological order. In both cases, the topological order on a single side surface breaks time-reversal symmetry, but appears with its time-reversal conjugate on alternating sides in a C_{2n}T preserving pattern. In the absence of the HOTI/HOTSC bulk, such a pattern necessarily involves gapless chiral modes on hinges between C_{2n}T-conjugate domains. However, using a combination of K-matrix and anyon condensation arguments, we show that on the boundary of a 3D HOTI/HOTSC these topological orders are fully gapped and hence "anomalous." Our results suggest that new patterns of surface and hinge states can be engineered by selectively introducing topological order only on specific surfaces.
Classification of symmetry-protected topological phases such as the topological superconductor (TSC) is well known in the free fermion picture. Recently, it was observed that including many-body interactions that preserves symmetry gives rise to novel surface phases and breaks down the classification scheme. A complete understanding of the microscopic theory for such phases is missing. This paper is an attempt to understand the interaction that gives rise to these phases in the case of the three-dimensional TSC. The authors build a coupled wire model Hamiltonian for the gapless surface and show that the addition of a time-reversal symmetric two-body interaction, constructed in this paper, opens an energy gap in the spectrum. They also show this gapped surface phase is topologically nontrivial and is classified differently compared to the noninteracting case.