The project has been dedicated to the studies of fractional topological phases occurring in the presence of strong electron-electron interactions. In particular, we studied the electrical conductance in single-mode quantum wires with Rashba spin-orbit interaction subjected to externally applied magnetic fields in the regime in which the ratio of spin-orbit momentum to the Fermi momentum is close to an odd integer, so that a combined effect of multielectron interaction and applied magnetic field leads to a partial gap in the spectrum. We studied how this partial gap manifests itself in the temperature dependence of the fractional conductance of the quantum wire and showed how the low-temperature fractional conductance can be affected by the finite length of the wire, by the properties of the contacts, and by a shift of the chemical potential, which takes the system away from the resonance condition. We proposed a tune-free scheme to realize Kramers pairs of Majorana bound states in recently discovered higher-order topological insulators (HOTIs): by bringing two hinges of a HOTI into the proximity of an s-wave superconductor, the competition between local and crossed Andreev pairing leads to the formation of Majorana Kramers pairs, when the latter pairing dominates over the former. We demonstrated that such a topological superconductivity is stabilized by moderate electron-electron interactions. In addition, we considered a Josephson junction bilayer consisting of two tunnel-coupled two-dimensional electron gas layers with Rashba spin-orbit interaction, proximitized by a top and bottom s-wave superconductor with phase difference $\phi$ close to $\pi$. In the presence of a finite weak in-plane Zeeman field, the bilayer can be driven into a second order topological superconducting phase, hosting two Majorana corner states. In the second year, we moved to the next stage of the project by including strong electron-electron interactions to generate fractional topological phases. In particular, we were interested in fractional higher-order topological insulators and superconductors, focusing on a system of weakly coupled Rashba nanowires in the strong spin-orbit interaction (SOI) regime. The nanowires were arranged into two tunnel-coupled layers proximitized by a top and bottom superconductor such that the superconducting phase difference between them is π. We showed that in such a system strong electron-electron interactions can stabilize a helical topological superconducting phase hosting Kramers partners of Z_2m parafermion edge modes, where m is an odd integer determined by the position of the chemical potential. Furthermore, upon turning on a weak in-plane magnetic field, the system is driven into a second-order topological superconducting phase hosting zero-energy Z_2m parafermion bound states localized at two opposite corners of a rectangular sample. We also considered a one-dimensional Rashba nanowire in which multiple Andreev bound states in the bulk of the nanowire form an Andreev band. We show that, under certain circumstances, this trivial Andreev band can produce an apparent closing and reopening signature of the bulk band gap in the nonlocal conductance of the nanowire. Furthermore, we show that the existence of the trivial bulk reopening signature in nonlocal conductance is essentially unaffected by the additional presence of trivial zero-bias peaks in the local conductance at either end of the nanowire. The simultaneous occurrence of a trivial bulk reopening signature and zero-bias peaks mimics the basic features required to pass the so-called “topological gap protocol.” Our results therefore provided a topologically trivial minimal model by which the applicability of this protocol can be benchmarked. This work was very well received by the community and was covered in the public media.
