Dynamics of the Schmid-Higgs Mode in $d$-wave superconductors

TL;DR

The paper analyzes the longitudinal Schmid–Higgs mode in a $d$-wave superconductor using a quasiclassical Eilenberger framework and Anderson pseudospin dynamics. It shows that the SH mode frequency is set by the anti-nodal pairing gap, $oxed{ω_{ ext{SH}}=2Δ_{ ext{an}}=2\,√2\,Δ}$, and that the oscillation amplitude decays as a power law with time, $oxed{ ext{(amplitude)} o 1/t^{2}}$, following a weak perturbation. The authors derive the SH susceptibility $oxed{ ext{χ}_{ ext{SH}}^{-1}(oldsymbol{Ω})}$ from the self-consistency equation and confirm their results via numerical time evolution of Anderson pseudospins, illustrating how nodal quasiparticles govern the damping. The work clarifies how the $d$-wave gap structure, particularly the anti-nodal region, dictates the SH mode energy and dissipation, with implications for non-equilibrium responses in cuprate-like superconductors.

Abstract

We study the dynamics of the longitudinal collective mode in an unconventional superconductor. For concreteness, we assume that the superconductor is described by a $d$-wave order parameter with $d_{x^2-y^2}$ symmetry. After the superconductor has been suddenly subjected to a perturbation at time $t=0$, the order parameter exhibits a peculiar oscillatory behavior, with the amplitude of the oscillations slowly decaying with time in a power-law fashion. Assuming that the initial perturbation is weak, we use a formalism based on quasi-classical approach to superconductivity to determine both the frequency of the oscillations as well as how fast these oscillations decay with time by evaluating the time dependence of the pairing susceptibility. We find that the frequency of the oscillations is given by twice the value of the pairing amplitude in the anti-nodal direction and its amplitude decays as $1/t^2$. The results are also verified by a direct calculation of the order parameter dynamics by numerically solving the equations of motion for the Anderson pseudospins.

Figures (3)

• Figure 1: Comparison between the real and imaginary parts of the function $\chi_{\textrm{SH}}(\Omega)$ for the $s$-wave and $d$-wave cases. For the $s$-wave case function $\mathrm{Im}[\chi_{\textrm{SH}}(\Omega)]$ exhibits a well-known Schmid-Higgs resonance at $\Omega=2\Delta$. In contrast for the $d$-wave pairing Schmid-Higgs susceptibility exhibits a maximum at frequency $\approx 2\sqrt{2}\Delta$.
• Figure 2: Time dependence of the Schmid-Higgs susceptibility computed from the Fourier transform of the function $\chi_{\mathrm{SH}}(\Omega)$ for the $s$-wave and $d$-wave cases. The dynamics in the $d$-wave case decays much faster. We found that at long times in the $d$-wave superconductor $\chi_{\mathrm{SH}}(t)\sim 1/t^2$. Inset: the plot of the results of the Fast Fourier Transform (FFT) showing that the $d$-wave pair susceptibility oscillates with the frequency $2\sqrt{2}\Delta$.
• Figure 3: Main panel: time dependence of the pairing amplitude following a small change of the pairing strength plotted in the units of equilibrium pairing amplitude $\Delta_0$. Inset: the fit of the $\Delta(t)$ at long times $t\Delta_0\gg 1$ showing that the oscillation amplitude decays as $\approx 1/t^2$.