A Time-Reversal Invariant Topological Phase at the Surface of a 3D Topological Insulator

TL;DR

The paper constructs a time-reversal and charge-conserving gapped surface topological order for a 3D fermionic topological insulator by condensing an $8\pi$ vortex in the Fu-Kane superconducting surface, yielding an Ising$\times$Z$_{8}^{(w)}$-type theory $X$ with $w=-\tfrac{1}{2}$ or $\tfrac{7}{2}$ that cannot arise in strictly two-dimensional systems. It analyzes the full anyon content (12 types), fusion rules, twist factors, and modular data (via a $\mathbb{Z}_2$-graded, spin modular framework), and demonstrates how time-reversal symmetry is realized in this nontrivial way. It also identifies the electron with the $(4,0)$ quasiparticle, derives charge-vorticity assignments through flux-threading arguments, and discusses edge theories and potential experimental realizations. The work connects to related Moore-Read-type physics and clarifies the role of vortex condensation in achieving symmetric surface topological order beyond the standard doubled or quantum-double constructions. Overall, it provides a principled route to time-reversal-invariant, charge-conserving surface orders on 3D TIs and lays groundwork for exploring similar phases with other non-Abelian anyons.

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 + i p_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.

Figures (4)

• Figure 1: (a) A toroidal slab of 3D TI has part of its upper surface in the gapped, time-reversal-invariant and charge-conserving topological phase $X$ while the rest of it is in the IQH state $M$. When flux $hc/e$ is threaded through the central hole, as shown, charge $e/2$ must accumulate (b) The lower surface of a slab of 3D TI is in the phase $M$ and the upper surface is in the phase $X$. In addition, we attach a 2D layer of electrons in the $\nu=1/2$ quantum Hall state $H$ of tightly-bound charge $-2e$ pairs in the $\nu_{\text{pair}}=1/8$ QH state of the bosonic pairs. The combined system is a 2D system in the Ising topological phase, and the excitation created by threading flux $hc/2e$ through the central tube is neutral.
• Figure 2: Quasiparticle types in theory $X$. The red hollow circles, blue solid circles, orange diamond and green diamond represent Abelian quasiparticles with $\theta=1$, $i$, $-1$, and $-i$ respectively. The hollow squares stand for the non-Abelian quasiparticles. For Abelian quasiparticles, the $x$ and $y$ coordinates of the solid circles indicate the charge and vorticity, respectively, of the quasiparticle. The non-Abelian quasiparticles do not have well-defined vorticity but have a well-defined charge represented by their $x$ coordinates.
• Figure 3: (a) A thin strip of $X$ separates $M_+$ and $M_-$, in which time-reversal symmetry is broken oppositely. As this strip becomes narrower, the two edges interact and form a single integer quantum Hall edge. (b) A thin strip of $M_+$ separates $X$ from $SC$. As this strip becomes narrower, the two edges interact and form a critical transverse field Ising model.
• Figure 4: When a ${\sigma_1}$ quasiparticle is taken around a ${\sigma_7}$ quasiparticle with which it fuses to $I_0$, the result is $R^{\sigma_1 \sigma_7}_{I_0}R^{\sigma_7 \sigma_1}_{I_0}$ when this is done along the black trajectory. When the trajectory is deformed to the blue one, then to the green one and, finally, to the orange one, the resulting phase is zero if the quasiparticle does not couple to the curvature.