Higher-order topological superconductivity in monolayer WTe$_2$ from repulsive interactions

Superconductivity has been experimentally observed in monolayer WTe2, which in-plane field measurements suggested are of spin-triplet nature. Furthermore, it has been proposed that with a $p$-wave pairing, the material is a second-order topological superconductor with a pair of Majorana zero modes at the corners of a finite sample. We show that for a repulsive on-site interaction and sizable Fermi surfaces, the desired p-wave state arises naturally due to the Kohn-Luttinger mechanism, and indeed a finite superconducting sample hosts corner Majorana zero modes. We study the behavior of the critical temperature in response to external in-plane magnetic fields. We find an enhancement to the critical temperature that depends on the direction of the magnetic field, which can be directly verified experimentally.