Dynamical generation of superconducting order of different symmetries in hexagonal lattices

The growth of superconducting order after an interaction quench in a hexagonal lattice is studied. The cases of both time-reversal (TR) preserving graphene, as well as the TR broken Haldane model are explored. Spin singlet superconducting order is studied where the $s$, $d+id$, and $d-id$ wave orders are the irreducible representations of the hexagonal lattice. For small quenches, the $d$-wave order parameter grows the fastest, a result also expected when the system is in thermal equilibrium. For the TR symmetry preserving case, the growth rate of the two $d$-wave orders is identical, while the TR-broken case prefers one of the chiral $d$-wave orders over the other, leading to a TR broken topological superconductor. As the interaction quench becomes larger, a smooth crossover is found where eventually the growth rate of the $s$-wave becomes the largest. Thus for large interaction quenches, the $s$-wave is preferred over the $d$-wave for both TR preserving and TR broken systems. This result is explained in terms of the high energy quasi-particles responsible for the dynamics as the interaction quench amplitude grows. The results are relevant for time-resolved measurements that can probe the symmetry of the superconducting fluctuations in a transient regime.