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arXiv:2409.11647 · Physical Review Letters

Spatially Resolved Dynamics of the Amplitude Schmid-Higgs Mode in Disordered Superconductors

P. A. Nosov, Elizaveta S. Andriyakhina, I. S. Burmistrov

Published Jul 14, 2025• 6 citations
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Abstract

We investigate the spatially resolved dynamics of the collective amplitude Schmid-Higgs (SH) mode in disordered s-wave superconductors and fermionic superfluids. By analyzing the analytic structure of the zero-temperature SH susceptibility in the complex frequency plane, we find that, when the coherence length greatly exceeds the mean free path, (i) the SH response at fixed wave vectors exhibits late-time oscillations decaying as 1/t^{2} with frequency 2Δ, where Δ is the superconducting gap; (ii) subdiffusive oscillations with a dynamical exponent z=4 emerge at late times and large distances; and (iii) spatial oscillations at a fixed frequency decay exponentially, with a period that diverges as the frequency approaches 2Δ from above. When the coherence length is comparable to the mean free path, additional exponentially decaying oscillations at fixed wave vectors appear with a frequency above 2Δ. Furthermore, we show that the SH mode induces an extra peak in the third-harmonic generation current at finite wave vectors. The frequency of this peak is shifted from the conventional resonance at Δ, thereby providing an unambiguous signature of order parameter amplitude dynamics.

DOI
10.1103/x12p-q7bj

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We investigate the spatially resolved dynamics of the collective amplitude Schmid-Higgs (SH) mode in disordered s-wave superconductors and fermionic superfluids. By analyzing the analytic structure of the zero-temperature SH susceptibility in the complex frequency plane, we find that, when the coherence length greatly exceeds the mean free path, (i) the SH response at fixed wave vectors exhibits late-time oscillations decaying as 1/t^{2} with frequency 2Δ, where Δ is the superconducting gap; (ii) subdiffusive oscillations with a dynamical exponent z=4 emerge at late times and large distances; and (iii) spatial oscillations at a fixed frequency decay exponentially, with a period that diverges as the frequency approaches 2Δ from above. When the coherence length is comparable to the mean free path, additional exponentially decaying oscillations at fixed wave vectors appear with a frequency above 2Δ. Furthermore, we show that the SH mode induces an extra peak in the third-harmonic generation current at finite wave vectors. The frequency of this peak is shifted from the conventional resonance at Δ, thereby providing an unambiguous signature of order parameter amplitude dynamics.
Symmetry-breaking phase transitions are central to our understanding of states of matter. When a continuous symmetry is spontaneously broken, new excitations appear that are tied to fluctuations of the order parameter. In superconductors and fermionic superfluids, the phase and amplitude can fluctuate independently, giving rise to two distinct collective branches. However, amplitude fluctuations are difficult to both generate and measure, as they do not couple directly to the density of fermions and have only been observed indirectly to date. Here, we excite amplitude oscillations in an atomic Fermi gas with resonant interactions by an interaction quench. Exploiting the sensitivity of Bragg spectroscopy to the amplitude of the order parameter, we measure the time-resolved response of the atom cloud, directly revealing amplitude oscillations at twice the frequency of the gap. The magnitude of the oscillatory response shows a strong temperature dependence, and the oscillations appear to decay faster than predicted by time-dependent Bardeen-Cooper-Schrieffer theory applied to our experimental setup.
The Higgs and Nambu-Goldstone modes are paradigmatic collective excitations in superconductors and superfluids. These modes are commonly analyzed within the pseudospin formulation of BCS theory, where the dynamics are obtained from the Heisenberg equations of motion for pseudospins. However, this construction becomes inconsistent when directly extended to non-Hermitian systems. In this work, we develop a consistent pseudospin framework for non-Hermitian fermionic superfluids based on the metricized formulation of non-Hermitian quantum mechanics. We apply this formalism to a driven non-Hermitian BCS-type Hamiltonian with complex pairing interaction and analyze its collective excitation spectrum. We find that, in addition to the conventional Higgs (amplitude) mode, the system hosts a novel phase mode that has no counterpart in Hermitian superfluids. Remarkably, this mode is gapped even in the absence of the Anderson-Higgs mechanism, as is the case in neutral superfluids. Furthermore, the resonance spectrum depends explicitly on the initial nongauge phase of the complex order parameter and the dynamical response remains finite at resonance, in contrast to the divergence characteristic of Hermitian systems. These resonances disappear upon the emergence of exceptional points.
We study collective amplitude modes of the superconducting order parameter in strongly coupled electron-phonon 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.
Using a state-of-the-art numerical scheme, we show that the Higgs mode under excitation exhibits chirped oscillations and exponential decay when fluctuations are included. This is in stark contrast to conventional BCS collisionless dynamics which predict power-law decay and the absence of chirping. The chirped amplitude mode enables us to determine the local modification of the effective potential even when the system is in a long-lived prethermal state. We then show that this chirped amplitude mode is an experimentally observable quantity since the photoinduced (super)current in pump-probe experiments serves as an efficient proxy for the order parameter dynamics, including the chirped dynamics. Our result is based on the attractive Hubbard model using dynamical mean-field theory within the symmetry-broken state after a excitation across the superconducting gap. Since the collective response involves long timescales, we extend the hierarchical low-rank compression method for nonequilibrium Green's functions to symmetry-broken states and show that it serves as an efficient representation despite long-lived memory kernels.
In quantum many-body systems with spontaneous breaking of continuous symmetries, Higgs modes emerge as collective amplitude oscillations of order parameters. Recently, Higgs mode has been observed in the ultracold Fermi gas. In the present paper, we use the time-dependent Bogoliubov-de Gennes equations to investigate Higgs amplitude oscillations of the superfluid order parameter in a Fermi gas induced by a rapid change of the s{\it s}s-wave scattering length. In particular, we investigate the Higgs mode with different values of the initial scattering length. We find that the energy of the Higgs mode coincides with the threshold energy of the pair-breaking excitation, and exponent of the power-low decay of the Higgs mode γγγ continuously changes between γ=−1/2γ=-1/2γ=−1/2 and γ=−3/2γ=-3/2γ=−3/2 through the Bardeen-Cooper-Schrieffer-Bose-Einstein condensation (BCS-BEC) crossover. Moreover, we propose the optimal ramp speed of the scattering length for observing the clearest Higgs oscillations.
We investigate the life cycle of the large amplitude Higgs mode in strongly interacting superfluid Fermi gas. Through numerical simulations with time-dependent density functional theory and the technique of the interaction quench, we verify the previous theoretical predictions on the mode's frequency. Next, we demonstrate that the mode is dynamically unstable against external perturbation and qualitatively examine the emerging state after the mode decays. The post-decay state is characterized by spatial fluctuations of the order parameter and density at scales comparable to the superfluid coherence length scale. We identify similarities with FFLO states, which become more prominent at higher dimensionalities and nonzero spin imbalances.