Collective modes in superconductors with competing s- and d-wave interactions

We calculate the collective mode spectrum in models of superconductors with attractive interactions in s-and d channels as a function of their relative strength, across a phase diagram that includes transition between s, s+id, and d-wave ground states. For one-band systems, we recover the largely known results for the phase, amplitude, and Bardasis-Schrieffer modes of pure s or d-states, and show how the well-defined Bardasis-Schrieffer mode softens near the s + id phase boundary and evolves in a characteristic manner through the s + id phase as a mixed symmetry mode. For two-band systems, we consider a model of hole-doped Fe-based superconductors, and find in the case of an s-wave ground state a well-defined Bardasis-Schrieffer mode below the lowest gap edge, as well as a second, damped mode of this type between the two gap energies. Both modes soften as the s + id phase is approached, and only a single "mixed-symmetry Bardasis-Schrieffer mode" below the pairbreaking continuum propagates in the s + id phase itself. These modes coexist with a damped Leggett mode with collective frequency between the two gap scales. In the pure d-state, no Bardasis-Schrieffer type s excitonic mode exists at low T. We briefly discuss Raman scattering experiments and how they can be used to identify an s + id state, and to track the evolution of competing s- and d- interactions in these systems.