Optically active Higgs and Leggett modes in multiband pair-density-wave superconductors with Lifshitz invariant

Lifshitz invariant is a symmetry invariant composed of multiple order parameters that contain a single spatial derivative in a Ginzburg-Landau (GL) free energy, which may induce a nonuniform configuration of the order parameters. In multiband superconductors, we find phase transitions from a uniform superconducting state to qualitatively distinct two pair-density-wave (PDW) states with small and large momenta $\boldsymbol{q}$ based on the GL theory. The former is induced by the Lifshitz invariant, while the latter originates from the drag effect. In the PDW states, the Higgs and Leggett modes (i.e., collective amplitude and relative phase oscillations of the order parameters) are shown to couple to electromagnetic fields linearly. We construct microscopic models of multiband superconductors with Lifshitz invariant that exhibit PDW states, and calculate the linear optical conductivity using the diagrammatic approach. We find optically active Higgs and Leggett modes in the small $\boldsymbol{q}$ PDW state, indicating that the PDW state is a suitable platform to explore collective modes of multiband superconductors in the linear response regime.