Flux sensitivity of quantum spin Hall rings

We analyse the periodicity of persistent currents in quantum spin Hall loops, partly covered with an $s$-wave superconductor, in the presence of a flux tube. Much like in normal (non-helical) metals, the periodicity of the single-particle spectrum goes from $\Phi_0 = h/e$ to $\Phi_0/2$ as the length of the superconductor is increased past the coherence length of the superconductor. We further analyze the periodicity of the persistent current, which is a many-body effect. Interestingly, time reversal symmetry and parity conservation can significantly change the period. We find a $2\Phi_0$-periodic persistent current in two distinct regimes, where one corresponds to a Josephson junction and the other one to an Aharonov-Bohm setup.