Interedge backscattering in time-reversal symmetric quantum spin Hall Josephson junctions

Using standard tight-binding methods, we investigate a novel backscattering mechanism taking place on quantum spin Hall N'SNSN' Josephson junctions in the presence of time-reversal symmetry. This extended geometry allows for the interplay between two types of Andreev bound states (ABS): the usual phase-dependent ABS localized at the edges of the central SNS junction \emph{and} phase-independent ABS localized at the edges of the N'S regions. Crucially, the latter arise at discrete energies $E_n$ and mediate a backscattering process between opposite edges on the SNS junction, yielding gap openings when both types of ABS are coherently coupled. In this scenario, a 4$\pi$-periodic ABS decouples from the rest of the 2$\pi$-periodic spectrum, yielding several observable consequences: Firstly, we show that the $4\pi$-periodic spectrum can be probed by means of the Shapiro experiment even in the presence of dynamical transitions between the ABS and the quasicontinuum. Secondly, the presence of this backscattering mechanism distorts the superconducting quantum interference (SQI) pattern within the length scale, determined by the ratio between $4\pi$- and $2\pi$-periodic supercurrent contributions. Finally, we propose to use a magnetic flux to tune $E_n$ to zero, resulting in the selective lifting of the fractional Josephson effect.