Non-Abelian Topological Order on the Surface of a 3D Topological Superconductor from an Exactly Solved Model

Three-dimensional (3D) topological superconductors (TScs) protected by time-reversal ($\mathcal{T}$) symmetry are characterized by gapless Majorana cones on their surface. Free-fermion phases with this symmetry (class DIII) are indexed by an integer $\nu$, of which $\nu =1$ is realized by the $B$ phase of superfluid ${}^3\mathrm{He}$. Previously, it was believed that the surface must be gapless unless time-reversal symmetry is broken. Here, we argue that a fully symmetric and gapped surface is possible in the presence of strong interactions, if a special type of topological order appears on the surface. The topological order realizes $\mathcal{T}$ symmetry in an anomalous way, one that is impossible to achieve in purely two dimensions. For odd $\nu \mathrm{TScs}$, the surface topological order must be non-Abelian. We propose the simplest non-Abelian topological order that contains electronlike excitations, $\mathrm{SO}(3{)}_6$, with four quasiparticles, as a candidate surface state. Remarkably, this theory has a hidden $\mathcal{T}$ invariance that, however, is broken in any two-dimensional realization. By explicitly constructing an exactly soluble Walker-Wang model, we show that it can be realized at the surface of a short-ranged entangled 3D fermionic phase protected by $\mathcal{T}$ symmetry, with bulk electrons transforming as Kramers pairs, i.e. ${\mathcal{T}}^2=-1$ under time reversal. We also propose an Abelian theory, the semion-fermion topological order, to realize an even $\nu \mathrm{TSc}$ surface, for which an explicit model is derived using a coupled-layer construction. We argue that this is related to the $\nu =2\mathrm{TSc}$, and we use this to build candidate surface topological orders for $\nu =4$ and $\nu =8\mathrm{TScs}$. The latter is equivalent to the three-fermion state, which is the surface topological order of a ${\mathbb{Z}}_2$ bosonic topological phase protected by $\mathcal{T}$ invariance. One particular consequence of this equivalence is that a $\nu =16\mathrm{TSc}$ admits a trivially gapped $\mathcal{T}$-symmetric surface.

Popular Summary

Topological superconductors are exotic electronic states of matter. They host fundamentally different electronic structures in their bulk and on their surfaces: The former is superconductorlike, and the latter is characterized by a single “Majorana cone”—electronic excitations that behave like Majorana fermions and whose energies form a gapless continuum. Such a gapless energy spectrum allows the surface electronic properties to be seamlessly varied, similar to the case of normal metals. On the other hand, an energy spectrum with a gap has its own use, similar to the case of semiconductors. It was previously thought that the only way to create a gap was to break one of the defining symmetries of the topological phase, the time-reversal symmetry. In this theoretical paper, we show that the Majorana cone can be “gapped” without breaking the time-reversal symmetry in the presence of strong electronic interactions, if a special type of topological order appears on the surface.

The topological order in question is defined by collective electronic excitations (or quasiparticles) that behave neither like bosons nor like fermions. In order to preserve the time-reversal symmetry, the topological order must behave in a certain way. We have proposed one of the simplest non-Abelian topological orders as the equivalent of a single Majorana cone, which can be realized on the surfaces of a three-dimensional system. Indeed, such new surface states may be more than just theoretical possibilities: The well-known superfluid phase of liquid ${}^3\mathrm{He}$, the so-called $B$ phase, is believed to be a three-dimensional topological superconductor.

This is a new and fundamental advance in the currently very active topical area of symmetry-protected topological states, in particular, in strongly interacting topological insulators and superconductors, and we expect it to inspire many further explorations.

Figure 1

Braiding statistics and fusion data for $\mathrm{SO}(3{)}_6$.

Figure 2

Graphical definition of the $F$ symbol, together with some identities satisfied by $F$.

Figure 3

Origin of ${\mathcal{T}}^2=-1$ in Walker-Wang models of $\mathrm{SO}(3{)}_6$ surface topological order. The additional phase factor shown in red depends on the orientation of the vertices and is required to ensure time-reversal symmetry. This leads to ${\mathcal{T}}^2=-1$ for fermions in this model.

Figure 4

Trivalent resolution of the cubic lattice [30].

Figure 5

Plaquette term in the Walker-Wang model (taken from Ref. 30).

Figure 6

Sample spin configuration in the decorated Walker-Wang model. Black dots represent spin $1/2$’s, and blue ellipses represent spin singlets. The leftmost vertex has a counterclockwise $(e,s,\tilde{s})$ ordering of labels and so prefers a down spin, whereas at the rightmost vertex, the counterclockwise label ordering is $(e,\tilde{s},s)$ and an up spin is preferred (red arrow).

Figure 7

Coupled-layer construction of semion-fermion topological order on the surface of a 3D topological superconductor (with even $\nu$). Each layer has a double-semion model and a ${\mathcal{Z}}_2$ gauge theory. The composite excitations enclosed by the brackets can be condensed simultaneously, after which the only deconfined excitations in the bulk are electrons, $c=b\psi$, shown by the dashed green boxes. At the surface, two additional excitations emerge—shown by the dashed green lines—which are exchanged by $\mathcal{T}$. A key ingredient is that the two bosons (electric and magnetic charges) of the ${\mathcal{Z}}_2$ gauge theory are exchanged by $\mathcal{T}$, which requires ${\mathcal{T}}^2=-1$ acting on electrons.