Pairing Symmetry and Projective Symmetry Groups for Fermions

Determining the pairing symmetry in a superconductor is a question of both conceptual and practical importance. In this work, we elucidate the connection between pairing symmetry of a superconductor and the fermion projective symmetry group (PSG). The normal state before the superconducting pairing formation has the symmetry X´U(1), where X denotes the crystalline symmetry. The formation of superconducting pairing breaks the U(1) charge conservation down to $Z_2^f$ fermion parity symmetry. The correct symmetry group for fermions in the superconducting phase is therefore a central extension of X by $Z_2^f$, mathematically described by the fermion PSG. It turns out that the representation of the pairing order parameter is intimately related to the PSGs of the fermions. We studied in detail the correspondence between fermion PSGs and the superconductor pairing symmetry for all point group symmetries. The fermion PSG constrains many other properties of the system, such as selection rules in Raman spectroscopy, classification of topological superconductors, and the presence/absence of Majorana zero modes at vortex cores. Our work provides a general framework to determine pairing symmetry through the study of fermion PSGs.