Multiple amplitude modes in strongly coupled phonon-mediated superconductors

Abstract

We study collective amplitude modes of the superconducting order parameter in strongly coupled electronphonon systems described by the Holstein model using the nonequilibrium dynamical mean-field theory with the self-consistent Migdal approximation as an impurity solver. The frequency of the Higgs amplitude mode is found to coincide with the superconducting gap even in the strongly coupled (beyond BCS) regime. Besides the Higgs mode, we find another collective mode involving the dynamics of both the phonons and the superconducting order parameter. The frequency of this mode, higher than twice the renormalized phonon frequency in the superconducting phase, is shown to reflect a strong electron-mediated phonon-phonon interaction. Both of collective modes are predicted to contribute to time-resolved photoemission spectra after a strong laser pump as vertex corrections to produce resonance peaks, which allows one to distinguish them from quasiparticle excitations.

Nonlinear light–Higgs coupling in superconductors beyond BCS: Effects of the retarded phonon-mediated interaction

“…The result is in agreement with the one previously reported. 48,51 The resonance peak is produced by the vertex correction, which is immediately confirmed by the comparison to the bare susceptibility. This suggests that the peak in χ R pair (ω) represents the collective oscillation of the pairing amplitude with the frequency 2∆, which can be identified as the Higgs amplitude mode.…”

“…The coincidence of the single-particle and two-particle gaps (up to the factor of 2) holds beyond the BCS approximation, as observed in the previous study. 48,51 The results of the THG susceptibility |χ| 2 (proportional to the THG intensity observed in experiments) calculated by the dotted DMFT are plotted in Fig. 6.…”

“…5, we show the single-particle spectrum A(ω) = −Im G R 11 (ω)/π (red curve) along with the dynamical pair susceptibility −Im χ R pair (ω) (50) (blue with the dots), with the latter calculated by the dotted DMFT. Previously, the dynamical pair susceptibility has been evaluated from the real-time simulation of the nonequilibrium DMFT, 48 which is one way to avoid solving the complicated Bethe-Salpeter equation for the vertex correction. Here the dotted DMFT serves as an alternative efficient method.…”

“…In particular, if we expand the dotted self-energy only with G and D 0 , it has a ladder structure, which represents the contribution from the amplitude mode. 48 The technique can be applied to any models in principle, as far as the diagrammatic expression for the impurity problem is given. To see how it works further, let us take another prototypical example, the Hubbard model,…”

“…It may contain a derivative of the selfenergy, which can be evaluated by the corresponding dotted DMFT, where the DMFT self-consistency equations are differentiated with respect to the external field. For example, the dynamical pair susceptibility, 40,48 …”