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Inherited topological superconductivity in two-dimensional Dirac semimetals

Under what conditions does a superconductor inherit topologically protected nodes from its parent normal state? In the context of Weyl semimetals with broken time-reversal symmetry, the pairing order parameter is classified by monopole harmonics and necessarily nodal [Li and Haldane, Phys. Rev. Lett., 120, 067003 (2018)]. Here, we show that a similar conclusion could also apply to 2D Dirac semimetals, although the conditions for the existence of nodes are more complex, depending on the pairing matrix structure in the valley and sublattice space. We analytically and numerically analyze the Bogoliubov-de-Gennes quasi-particle spectra for Dirac systems based on the monolayer as well as twisted bilayer graphene. We find that in the cases of intra-valley intra-sublattice pairing, and inter-valley inter-sublattice pairing, the point nodes in the BdG spectrum (which are inherited from the Dirac cone in the normal state) are protected by a 1D winding number. The nodal structure of the superconductivity is confirmed using tight-binding models of monolayer and twisted bilayer graphene. Notably, the BdG spectrum is nodal even with a momentum-independent "bare" pairing, which, however, acquires momentum-dependence and point nodes upon projection to the Bloch states on the topologically nontrivial Fermi surface, similar in spirit to the Li--Haldane monopole superconductor and the Fu--Kane proximity-induced superconductor on the surface of a topological insulator.