Engineering Higgs dynamics by spectral singularities

We generalize the dynamical phase diagram of a Bardeen-Cooper-Schrieffer condensate, considering attractive to repulsive, i.e., critical quenches (CQs) and a nonconstant density of states (DOS). We show that different synchronized Higgs dynamical phases can be stabilized, associated with singularities in the DOS and different quench protocols. In particular, the CQ can stabilize an overlooked high-frequency Higgs dynamical phase related to the upper edge of the fermionic band. For a compensated Dirac system we find a Dirac-Higgs mode associated with the cusp singularity at the Fermi level, and we show that synchronized phases become more pervasive across the phase diagram. The relevance of these remarkable phenomena and their realization in ensembles of fermionic cold atoms confined in optical lattices is also discussed.