We investigate the topological degeneracy that can be realized in Abelian fractional quantum spin Hall states with multiply connected gapped boundaries. Such a topological degeneracy (also dubbed as "boundary degeneracy") does not require superconducting proximity effect and can be created by simply applying a depletion gate to the quantum spin Hall material and using a generic spin-mixing term (e.g., due to backscattering) to gap out the edge modes. We construct an exactly soluble microscopic model manifesting this topological degeneracy and solve it using the recently developed technique [S. Ganeshan and M. Levin, Phys. Rev. B 93, 075118 (2016)]. The corresponding string operators spanning this degeneracy are explicitly calculated. It is argued that the proposed scheme is experimentally reasonable.

1 aGaneshan, Sriram1 aGorshkov, Alexey, V.1 aGurarie, Victor1 aGalitski, Victor, M. uhttp://journals.aps.org/prb/abstract/10.1103/PhysRevB.95.04530900913nas a2200145 4500008004100000245004700041210004700088260001300135490000700148520050500155100002600660700001900686700002500705856003700730 2011 eng d00aInterferometry with Synthetic Gauge Fields0 aInterferometry with Synthetic Gauge Fields c2011/3/30 v833 aWe propose a compact atom interferometry scheme for measuring weak, time-dependent accelerations. Our proposal uses an ensemble of dilute trapped bosons with two internal states that couple to a synthetic gauge field with opposite charges. The trapped gauge field couples spin to momentum to allow time dependent accelerations to be continuously imparted on the internal states. We generalize this system to reduce noise and estimate the sensitivity of such a system to be S~10^-7 m / s^2 / Hz^1/2. 1 aAnderson, Brandon, M.1 aTaylor, J., M.1 aGalitski, Victor, M. uhttp://arxiv.org/abs/1008.3910v2