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Condensed Matter

Undamped Higgs Modes in Strongly Interacting Superconductors

J. Lorenzana, G. Seibold

Published Sep 30, 2024• 1 citation
Publisher

Abstract

In superconductors gauge U(1) symmetry is spontaneously broken. According to Goldstone theorem this breaking of a continuous symmetry establishes the existence of the Bogoliubov phase mode while the gauge-invariant response also includes the amplitude fluctuations of the order parameter. The latter, which are also termed ’Higgs’ modes in analogy with the standard model, appear at the energy of the spectral gap 2Δ when the superconducting ground state is evaluated within the weak coupling BCS theory and are therefore damped. Previously we have shown that in the time-dependent Gutzwiller approximation (TDGA) Higgs modes appear inside the gap with a finite binding energy relative to the quasiparticle continuum. Here we show that the binding energy of the Higgs mode becomes exponentially small in the weak coupling limit converging to the BCS solution. On the other hand, well-defined undamped amplitude modes exist in strongly coupled superconductors when the interaction energy becomes of the order of the bandwidth.

DOI
10.3390/condmat9040038

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In superconductors gauge U(1) symmetry is spontaneously broken. According to Goldstone theorem this breaking of a continuous symmetry establishes the existence of the Bogoliubov phase mode while the gauge-invariant response also includes the amplitude fluctuations of the order parameter. The latter, which are also termed ’Higgs’ modes in analogy with the standard model, appear at the energy of the spectral gap 2Δ when the superconducting ground state is evaluated within the weak coupling BCS theory and are therefore damped. Previously we have shown that in the time-dependent Gutzwiller approximation (TDGA) Higgs modes appear inside the gap with a finite binding energy relative to the quasiparticle continuum. Here we show that the binding energy of the Higgs mode becomes exponentially small in the weak coupling limit converging to the BCS solution. On the other hand, well-defined undamped amplitude modes exist in strongly coupled superconductors when the interaction energy becomes of the order of the bandwidth.
We investigate order parameter fluctuations in the Hubbard model within a time-dependent Gutzwiller approach. While in the weak coupling limit we find that the amplitude fluctuations are short-lived due to a degeneracy with the energy of the edge of the quasiparticle continua (and in agreement with Hartree-Fock+RPA theory), these are shifted below the edge upon increasing the interaction. Our calculations therefore predict undamped amplitude (Higgs) oscillations of the order parameter in strongly coupled superconductors, cold atomic fermion condensates, and strongly interacting charge- and spin-density wave systems. We propose an experimental realization for the detection of the spin-type Higgs mode in undoped cuprates and related materials where, due to the Dzyaloshinsky-Moriya interaction, it can couple to an out-of-plane ferromagnetic excitation that is visible via the Faraday effect.
Collective modes in a superconductor correspond to fluctuations in the amplitude (Higgs mode) or the phase (Goldstone mode) of the order parameter. Starting from the BCS Hamiltonian, we derive a microscopic Hamiltonian for the coupled Higgs-Goldstone-electron system that contains all interactions among these degrees of freedom. The Higgs subsystem is a weakly interacting Bose gas: Higgs modes experience a mutual attraction mediated by the virtual exchange of Bogoliubov quasi-particle pairs. They also interact with external electromagnetic-fields with a coupling constant that we derive explicitly. We use this theory to examine whether an optically pumped gas of Higgs modes can undergo Bose-Einstein condensation. We identify a density window, bounded below by the equilibrium condensation criterion and above by collapse due to the attractive interaction, in which condensation can occur and we derive the expected density of Higgs-modes due to external pumping. This pumping depends crucially on non-parabolic corrections to the electron dispersion around the Fermi-energy. We show that the pumped mode density is compatible with condensation for realistic material parameters. We suggest possible experimental signatures of this non-equilibrium condensate.
The amplitude mode of a fermionic superfluid, analogous to the Higgs Boson, becomes undamped in the strong coupling regime when its frequency is pushed inside the BCS energy gap. We argue that this is the case in cold gases due to the energy dispersion and nonlocality of the pairing interaction, and propose to use the Feshbach resonance regime for parametric excitation of this mode. The results presented for the BCS pairing dynamics indicate that even weak dispersion suppresses dephasing and gives rise to persistent oscillations. The frequency of oscillations extracted from our simulation of the BCS dynamics agrees with the prediction of the many-body theory.
We study the effect of strong electron-phonon interactions on the damping of the Higgs amplitude mode in superconductors by means of nonequilibrium dynamical mean-field simulations of the Holstein model. In contrast to the BCS dynamics, we find that the damping of the Higgs mode strongly depends on the temperature, becoming faster as the system approaches the transition temperature. The damping at low temperatures is well described by a power law, while near the transition temperature the damping shows exponential-like behavior. We explain this crossover in terms of a temperature-dependent quasiparticle lifetime caused by the strong electron-phonon coupling, which smears the superconducting gap edge and makes the relaxation of the Higgs mode into quasiparticles more efficient at elevated temperatures. We also reveal that the phonon dynamics can soften the Higgs mode, which results in a slower damping.
By studying the two-dimensional Su-Schrieffer-Heeger-Bose-Hubbard model, we show the existence of topological Higgs amplitude modes in the strongly interacting superfluid phase. Using the slave boson approach, we find that, in the large filling limit, the Higgs excitations and the Goldstone excitations above the ground state are well decoupled, and both of them exhibit nontrivial topology inherited from the underlying noninteracting bands. At finite fillings, they become coupled at high energies; nevertheless, the topology of these modes remains unchanged. Based on an effective action analysis, we further provide a universal physical picture for both their topological and phase-amplitude character at both infinite and finite fillings in a unified way. The discovery of topological Higgs amplitude modes in this paper opens the path to investigations in various systems, such as superconductors and quantum magnets.
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.