Time-reversal symmetry breaking, collective modes, and Raman spectrum in pair-density-wave states

Inspired by empirical evidence of the existence of pair-density-wave (PDW) order in certain underdoped cuprates, we investigate the collective modes in systems with unidirectional PDW order with momenta ± Q and a d-wave form-factor with special focus on the amplitude (Higgs) modes. In the pure PDW state, there are two overdamped Higgs modes. We show that a phase with co-existing PDW and uniform (d-wave) superconducting (SC) order, PDW/SC, spontaneously breaks time-reversal symmetry—and thus is distinct from a simpler phase, SC/CDW, with coexisting SC and charge-density-wave (CDW) order. The PDW/SC phase exhibits three Higgs modes, one of which is sharply peaked and is predominantly a PDW fluctuation, symmetric between Q and −Q, whose damping rate is strongly reduced by SC. This sharp mode should be visible in Raman experiments.