Ground-State Degeneracy of Topological Phases on Open Surfaces

We relate the ground state degeneracy of a non-Abelian topological phase on a surface with boundaries to the anyon condensates that break the topological phase into a trivial phase. Specifically, we propose that gapped boundary conditions of the surface are in one-to-one correspondence with the sets of condensates, each being able to completely break the phase, and we substantiate this by examples. The ground state degeneracy resulting from a particular boundary condition coincides with the number of confined topological sectors due to the corresponding condensation. These lead to a generalization of the Laughlin-Tao-Wu charge-pumping argument for Abelian fractional quantum Hall states to encompass non-Abelian topological phases, in the sense that an anyon loop of a confined anyon winding a nontrivial cycle can pump a condensed anyon from one boundary to another. Such generalized pumping may find applications in quantum control of anyons, eventually realizing topological quantum computation.

Figure 1

Ground state basis specified by a conserved anyon line $m$ winding the cylinder where the boundaries are characterized by the $L_e$ condensate. A unit of $e$ is transferred across the boundaries via the LTW charge pumping mechanism if the unit $m$ line can be changed to two units adiabatically.

Figure 2

(a) $F\times \overline{F}$ on a cylinder with both boundaries characterized by the Lagrangian subset $\{1,\tau \overline{\tau }\}$. (b) Single Fibonacci phase on a torus. The two systems are equivalent.

Figure 3

(a) An open surface with $M>3$ boundaries. (b) A genus-$(M-1)$ torus.

Figure 4

${\mathbb{Z}}_2^3$ twisted quantum double on a cylinder with the left boundary characterized by electric condensation $L_E$ and the right one by dyonic condensation $L_D$. $\text{GSD}=|T_E\cap T_D|$.