Recently, a lot of interest has been raised by the generalization of conventional TIs/TSCs to so-called higher order TIs/TSCs. While a conventional d-dimensional TI/TSC exhibits (d − 1)-dimensional gapless boundary modes, a d-dimensional n-th order TI/TSC hosts gapless modes at its (d − n)-dimensional boundaries. In my talk, I will consider 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 φ close to π [1-3]. 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 (MCSs). If φ=π, in a rectangular geometry, these zero-energy bound states are located at two opposite corners determined by the direction of the Zeeman field. If the phase difference φ deviates from π by a critical value, one of the two MCSs gets relocated to an adjacent corner. As the phase difference φ increases further, the system becomes trivially gapped. The obtained MCSs are robust against static and magnetic disorder.
In the second part of my talk, I will switch from non-interacting systems [4,5], in which one neglects effects of strong electron-electron interactions, to interacting systems and, thus, to exotic fractional phases. I will show that this is indeed possible and explicitly construct a two-dimensional (2D) fractional second-order TSC. I will consider a system of weakly coupled Rashba nanowires in the strong spin-orbit interaction (SOI) regime. The nanowires are arranged into two tunnel-coupled layers proximitized by a top and bottom superconductor such that the superconducting phase difference between them is π. 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.
References Y. Volpez, D. Loss, and J. Klinovaja, Phys. Rev. Lett. 122,126402 (2019)
 K. Plekhanov, M. Thakurathi, D. Loss, and J. Klinovaja, Phys. Rev. Research 1, 032013(R) (2019)
 K. Plekhanov, N. Müller, Y. Volpez, D. M. Kennes, H. Schoeller, D. Loss, and J. Klinovaja, arXiv:2008.03611
 K. Laubscher, D. Loss, and J. Klinovaja, Phys. Rev. Research 1, 032017(R) (2019)
 K. Laubscher, D. Loss, and J. Klinovaja, Phys. Rev. Research 2, 013330 (2020)