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, $\text{Moore-Read}\times \mathrm{U}{\left(1\right)}_{-2}$. The Moore-Read sector supports non-Abelian charge $1/4$ anyons, while the Abelian, $\mathrm{U}{\left(1\right)}_{-2}$, (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 $X{Y}^{*}$ 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 $4\pi (2hc/e)$ 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.

Figure 1

An ETI slab (pink) with the top surface in a $\mathcal{T}$-preserving superconducting phase (blue) and the bottom surface in a $\mathcal{T}$-broken insulating phase with ${\sigma }_{xy}=1/2$ (orange). The entire slab viewed as a 2d system is a $p_x+i{p}_y$ superconductor, as can be deduced from the presence of the chiral Majorana edge mode $f$. A vortex on the superconducting surface will have its magnetic field (red) spread into the bulk of the slab. As a whole, the flux-tube piercing the slab has Ising statistics (see Table 3). The intrinsic vortex statistics associated with the $\mathcal{T}$-invariant superconducting surface can be deduced by subtracting out the contribution of the bottom surface, which gives the restricted $\text{Ising}\times \mathrm{U}{\left(1\right)}_{-8}$ vortex theory in Table 5.

Figure 2

A disk of the 2d topological superconductor in class D III surrounded by an annulus of the $\mathcal{T}$-invariant $T_{24}$ phase. The latter can be obtained using a slab of an ETI with symmetry-preserving $T_{24}$ topological order on the top surface and the superconducting phase on the bottom surface. The disk + annulus system exhibits the phenomenon of “weak symmetry breaking” first introduced in Refs. [55, 56]: both edges of the annulus are $T$-preserving and gapped, yet the entire annulus breaks $\mathcal{T}$. The order parameter for $\mathcal{T}$ breaking is provided by an operator, which tunnels the boson $e^{2i\varphi }$ across the annulus.