Weak-Pairing Higher Order Topological Superconductors*

We consider a new class of superconductors, second-order topological superconductors (SOTSC), that have gapped, topological surfaces and gapless Majorana modes on lower-dimensional boundaries: corners of a 2D system or hinges for a 3D system. We propose two general scenarios that SOTSC can be realized spontaneously with weak-pairing instabilities. First, we show that p-wave pairing in a Dirac semimetal in 2D with four mirror symmetric Dirac nodes realizes SOTSC. Second, we show that p+id pairing on an ordinary Fermi surface realizes SOTSC as well. We show that these exotic states can be intrinsically realized in a metallic system with electronic interactions. In the latter case we find that the topological invariants describing the system can be written using simple formulae involving only the low-energy properties of the Fermi surfaces and superconducting pairing, and we also show it can be induced by proximity effect in a heterostructure of cuprate and superconductor.