We analyze the spectrum of collective modes in a superconductor, in which pairing is mediated by long-range nematic fluctuations. Previous experimental and theoretical studies have found that a superconducting gap in such a system is highly anisotropic and at any finite $T<T_c$ vanishes on four arcs of the Fermi surface even when the pairing symmetry is $s$ wave ($s^{+-}$ between hole and electron pockets). We derive the expression for the pair susceptibility $\chi (\mathbf{q},\mathrm{\Omega })$ at finite momentum $\mathbf{q}$ and frequency $\mathrm{\Omega }$ deep in the superconducting phase. We analyze the spectral function, $\text{Im}\chi (\mathbf{q},\mathrm{\Omega })$, and its pole structure in transverse (phase) and longitudinal (amplitude) channels and compare the results with those of a conventional $s$-wave superconductor. We find that the analytic structure of the pair susceptibility in both channels is qualitatively distinct from that in a BCS superconductor. This gives rise to a highly unconventional dispersion of phase and amplitude collective modes.

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• Nematic order
• Pairing mechanisms
• Quasiparticles & collective excitations
• Superconducting gap
• Unconventional superconductors
• Green's function methods
• Random phase approximation