Anomalous Quasiparticle Symmetries and Non-Abelian Defects on Symmetrically Gapped Surfaces of Weak Topological Insulators

We show that boundaries of 3D weak topological insulators can become gapped by strong interactions while preserving all symmetries, leading to Abelian surface topological order. The anomalous nature of weak topological insulator surfaces manifests itself in a nontrivial action of symmetries on the quasiparticles; most strikingly, translations change the anyon types in a manner impossible in strictly 2D systems with the same symmetry. As a further consequence, screw dislocations form non-Abelian defects that trap ${\mathbb{Z}}_4$ parafermion zero modes.

Figure 1

(a) A weak topological insulator built from quantum spin Hall layers. Interlayer tunneling yields two symmetry-protected surface Dirac points at momenta $(q_x,q_y)=(0,0)$ and $(0,\pi )$. (b) A thickened torus of a weak topological insulator with a symmetric topologically ordered interior and a gapless exterior. (c) Two-dimensional limit where the thickness shrinks to zero. The topological order must be anomalous; otherwise, one is left with an impossible 2D band structure. This very general argument applies broadly to 3D symmetry-protected topological phases.

Figure 2

(a) A weak topological insulator surface dressed with 2D topologically ordered plates. (b) The limit of a single QSH layer which provides the setup for discussing weak symmetry breaking.

Figure 3

(a) Dragging an anyon across plates leaves behind “invisible” operators at the interface that get absorbed into a condensate. The condensates allow the $a$ quasiparticle, carrying charge $e/2$, to pass freely between plates while the neutral $d$ quasiparticle acquires a neutral fermion and thus changes anyon type. (b) A weak topological insulator surface with a screw dislocation. Upon encircling the dislocation, $\tilde{d}$ anyons acquire a neutral fermion, indicating a zero mode bound to the defect.