Schmid-Higgs Mode in the Presence of Pair-Breaking Interactions

TL;DR

The paper investigates the Schmid-Higgs amplitude mode in superconductors with pair-breaking, showing that even when the quasiparticle gap ε_g lies inside the continuum, the SH mode retains an algebraic, non-damped-like time evolution after a sudden perturbation, with oscillations at frequency 2 ε_g and a t^{-2} decay modulated by γ-dependent factors. Using a real-time Keldysh sigma-model and a generalized Usadel equation, the authors derive the longitudinal susceptibility χ_SH and reveal a Lorentzian-like two-photon absorption peak at ω_res within the continuum. The key contributions include the identification of a robust algebraic decay in the presence of pair-breaking, the characterization of ω_res and its γ-dependence, and the delineation of the gapless regime ε_g=0, where exponential damping emerges, with implications for finite-temperature and finite-q extensions. These results reinforce the resilience of the Higgs-like mode under weak time-reversal-symmetry-breaking perturbations and provide concrete predictions for nonlinear optical responses and time-domain dynamics in disordered superconductors.

Abstract

Collective modes in superconductors provided the first realization of the Higgs mechanism. The transverse Goldstone mode acquires a gap (i.e. a mass) when it hybridizes with the electromagnetic gauge field. The longitudinal Schmid-Higgs mode, on the other hand, is always massive. In conventional BCS theory, its gap is exactly $2Δ$, coinciding with the excitation threshold for quasiparticles. Being situated right at the edge of the continuum spectrum it gives rise to peculiar dynamics for the Schmid-Higgs mode. For instance, when suddenly excited at $t=0$, it exhibits algebraically decaying oscillations of the form $\sim \sin(2Δt)/{t}^{1/2}$. In this study, we explore the behavior of Schmid-Higgs oscillations in the presence of pair-breaking mechanisms, such as magnetic impurities or in-plane magnetic fields. These processes suppress the quasiparticle excitation threshold down to $2\varepsilon_g < 2Δ$, potentially placing the longitudinal mode within the continuum spectrum. Despite this, we show that the algebraically decaying oscillations persist, taking the form $\sim \sin(2\varepsilon_g t)/t^2$. The Schmid-Higgs mode becomes truly overdamped and exponentially decaying only in the gapless superconductors with $\varepsilon_g=0$.

Figures (5)

• Figure 1: Dependence of the real and imaginary part of the function $\eta_\epsilon^R$, Eq. (\ref{['etaRA']}), as a function of energy for various values of the pair breaking parameter corresponding to different values of $\varepsilon_g/\Delta$.
• Figure 2: Inverse longitudinal susceptibility $\chi^{-1}_{\textrm{SH}}(\omega,\mathbf q=0)$, as a function of frequency for various values of the ratio $\varepsilon_g/\Delta$, in the limit $T\to 0$.
• Figure 3: Dependence of the real and imaginary parts of the amplitude mode susceptibility $\chi_{\textrm{SH}}(\omega,\mathbf q=0)$, as a function of energy for various values of the ratio $\varepsilon_g/\Delta$ in the limit $T\to0$.
• Figure 4: Longitudinal mode response on a sudden quench perturbation at $t=0$, evaluated for different values of the dimensionless parameter $\gamma=1/\tau_s\varepsilon_g$. Here $\Delta_0$ is the superconducting pairing gap for $\tau_s\to\infty$. At long times, such that $t\gg\tau_s \gamma^{1/3}$, the susceptibility asymptotes to zero as $\sim 1/t^2$.
• Figure 5: Time evolution of the Schmid-Higgs susceptibility evaluated for $\tau_s\Delta\approx1$ when a superconductor is close to the gapless state $\varepsilon_g/\Delta=0$ (inset). Here $\Delta_0$ is the superconducting pairing gap for $\tau_s\to\infty$. At long times, such that $t\gg\tau_s$, the susceptibility asymptotes to zero as $1/t^2$. Inset: dependence of the quasiparticle (half) gap, $\varepsilon_g$, and the order parameter, $\Delta$, on the pair-breaking rate, $1/\tau_s$. The gapless superconductivity regime is seen for $0.228 < (2\tau_s\Delta_0)^{-1}< 0.25$.