Superconductivity mediated by nematic fluctuations -- the dispersion of collective modes

Abstract

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 the 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 $\chi (\mathbf{q},\Omega )$ at finite momentum $\mathbf{q}$ and frequency $\Omega$ deep in the superconducting phase. We analyze the spectral function, $\mathrm{Im}\chi (\mathbf{q},\Omega )$, and its pole structure in the 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.