Superfluid Stiffness of Systems with Bogoliubov Fermi Surfaces

Bogoliubov Fermi surfaces are topologically protected regions of zero energy excitations in a superconductor whose dimension equals that of the underlying normal state Fermi surface. Hamiltonians exhibiting this "ultranodal" phase are regarded as unphysical if superfluid stiffness is negative. In this work we show that superfluid stiffness is positive within large areas of the parameter space of our previously proposed model Hamiltonian, confirming that the proposed model has stable ultranodal phase. We further investigate the the free energy expansion beyond Gaussian levell, and investigate the possibility that the instability in the case of negative stiffness can be to pair density wave order.