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arXiv:1306.3286 · Physical Review B

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

Max A. Metlitski, C. L. Kane, Matthew P. A. Fisher

Published Sep 8, 2015• 154 citations
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Abstract

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 $X{Y}^{*}$ 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.

DOI
10.1103/physrevb.92.125111

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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\mathcal{T}T or U(1)\text{U}(1)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\mathcal{T}T symmetry. However, by constructing an exactly soluble 3D lattice model, we show they can be realized as T\mathcal{T}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\mathcal{T}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\mathcal{T}T-symmetric surface state.
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.
It is well known that one signature of the three-dimensional electron topological insulator is the Witten effect: if the system is coupled to a compact electromagnetic gauge field, a monopole in the bulk acquires a half-odd-integer polarization charge. In the present work, we propose a corresponding phenomenon for the topological insulator of bosons in three dimensions protected by particle number conservation and time-reversal symmetry. We claim that although a monopole inside a topological insulator of bosons can remain electrically neutral, its statistics are transmuted from bosonic to fermionic. We demonstrate that this ``statistical Witten effect'' directly implies that if the surface of the topological insulator is neither gapless, nor spontaneously breaks the symmetry, it necessarily supports an intrinsic two-dimensional topological order. Moreover, the surface properties can not be fully realized in a purely two-dimensional system. We also confirm that the surface phases of the bosonic topological insulator proposed by Vishwanath and Senthil [Phys. Rev. X 3, 011016 (2013)] provide a consistent termination of a bulk exhibiting the statistical Witten effect. In a forthcoming paper, we will provide an explicit field-theoretic, lattice-regularized construction of the three-dimensional topological insulator of bosons, employing a parton decomposition and subsequent condensation of parton-monopole composites.
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.
The surface electronic structure of a three-dimensional topological superconductor is characterized by a Majorana cone---Majorana-fermion-like electronic excitations whose energies form a gapless continuum, rendering the surface conducting. It was believed that a gap in that electronic structure could only be created if time-reversal symmetry were broken. Theorists now show that, in the presence of strong electronic interaction, ``gapping'' and time-reversal symmetry can coexist if the surface has a special type of electronic excitations that are ``anyons.''
Three dimensional topological superconductors with time reversal symmetry (class DIII) are indexed by an integer ννν, the number of surface Majorana cones, according to the free fermion classification. The superfluid B phase of He3^33 realizes the ν=1ν=1ν=1 phase. Recently, it has been argued that this classification is reduced in the presence of interactions to Z16_{16}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=2mν=2mν=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 Z8_88​. 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=±iT^2=\pm iT2=±i.