Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1511.06105 · Physical review. B./Physical review. B

Multiple amplitude modes in strongly coupled phonon-mediated superconductors

Yuta Murakami, Philipp Werner, Naoto Tsuji, Hideo Aoki

Published Mar 8, 2016• 46 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.93.094509

Related papers

Damping of the collective amplitude mode in superconductors with strong electron-phonon coupling

Yuta Murakami, Philipp Werner, Naoto Tsuji +1 more

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.

arXiv:1604.08073•Physical review. B./Physical review. B•Sep 12, 2016•18 citations

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

Naoto Tsuji, Yuta Murakami, Hideo Aoki

We study the contribution of the Higgs amplitude mode on the nonlinear optical response of superconductors beyond the BCS approximation by taking into account the retardation effect in the phonon-mediated attractive interaction. To evaluate the vertex correction in nonlinear optical susceptibilities that contains the effect of collective modes, we propose an efficient scheme which we call the ``dotted DMFT'' based on the nonequilibrium dynamical mean-field theory (nonequilibrium DMFT), to get around the difficulty of solving the Bethe-Salpeter equation and analytical continuation. The vertex correction is represented by the derivative of the self-energy with respect to the external driving field, which is self-consistently determined by the differentiated (``dotted'') DMFT equations. We apply the method to the Holstein model, a prototypical electron-phonon-coupled system, to calculate the susceptibility for the third-harmonic generation including the vertex correction. The results show that, in sharp contrast to the BCS theory, the Higgs mode can contribute to the third-harmonic generation for general polarization of the laser field with an order of magnitude comparable to the contribution from the pair breaking or charge density fluctuations. The physical origin is traced back to the nonlinear resonant light--Higgs coupling, which has been absent in the BCS approximation.

arXiv:1606.05024•Physical review. B./Physical review. B•Dec 29, 2016•57 citations

Collective modes in pumped unconventional superconductors with competing ground states

Marvin A. Müller, Pavel A. Volkov, Indranil Paul +1 more

Motivated by the recent development of terahertz pump-probe experiments, we investigate the short-time dynamics in superconductors with multiple attractive pairing channels. Studying a single-band square lattice model with a spin-spin interaction as an example, we find the signatures of collective excitations of the pairing symmetries (known as Bardasis-Schrieffer modes) as well as the order parameter amplitude (Higgs mode) in the short-time dynamics of the spectral gap and quasiparticle distribution after an excitation by a pump pulse. We show that the polarization and intensity of the pulse can be used to control the symmetry of the nonequilibrium state as well as frequencies and relative intensities of the contributions of different collective modes. We find particularly strong signatures of the Bardasis-Schrieffer mode in the dynamics of the quasiparticle distribution function. Our work shows the potential of modern ultrafast experiments to address the collective excitations in unconventional superconductors and highlights the importance of subdominant interactions for the nonequilibrium dynamics in these systems.

arXiv:1907.10464•Physical review. B./Physical review. B•Oct 15, 2019•24 citations

Collective mode across the BCS-BEC crossover in Holstein model

Tae-Ho Park, Han-Yong Choi

We investigate the emergence of the collective mode in the phonon spectra of the superconducting state within the Holstein model by varying the electron-phonon coupling. Using dynamical mean field theory combined with the numerical renormalization group technique, we calculate the phonon spectra. In the superconducting state with a pairing gap ( Δ P ), the peak position of the collective mode ( ω col ) evolves from the Bardeen-Cooper-Schrieffer (BCS) regime, manifesting near 2 Δ P and increasing with coupling, to the Bose-Einstein condensation (BEC) regime, where ω col decreases with increasing coupling. The decrease of ω col matches well with the reduction of superfluid stiffness, which originates from the increasing phase fluctuations of local pairs with coupling strength. In the crossover regime with intermediate coupling, ω col aligns with the soft phonon mode ( ω s ) of the normal state and decreases with increasing coupling when ω s < 2 Δ P . Additionally, comparing the collective mode weight to Δ P suggests that the collective mode predominantly stems from U(1) gauge symmetry breaking across all coupling strengths. Published by the American Physical Society 2024

arXiv:2411.14782•Physical Review Research•Dec 9, 2024•0 citations

Theory of light-enhanced phonon-mediated superconductivity

Michael A. Sentef, Alexander F. Kemper, Antoine Georges +1 more

We investigate the dynamics of a phonon-mediated superconductor driven out of equilibrium. The electronic hopping amplitude is ramped down in time, resulting in an increased electronic density of states. The dynamics of the coupled electron-phonon model is investigated by solving Migdal-Eliashberg equations for the double-time Keldysh Green's functions. The increase of the density of states near the Fermi level leads to an enhancement of superconductivity when the system thermalizes to the new state at the same temperature. We provide a time- and momentum-resolved view on this thermalization process and show that it involves fast processes associated with single-particle scattering and much slower dynamics associated with the superconducting order parameter. The importance of electron-phonon coupling for the rapid enhancement and the efficient thermalization of superconductivity is demonstrated, and the results are compared to a BCS time-dependent mean-field approximation.

arXiv:1505.07575•Physical review. B./Physical review. B•Apr 8, 2016•159 citations

Ordered phases in the Holstein-Hubbard model: Interplay of strong Coulomb interaction and electron-phonon coupling

Yuta Murakami, Philipp Werner, Naoto Tsuji +1 more

We study the Holstein-Hubbard model at half filling to explore ordered phases including superconductivity (SC), antiferromagnetism (AF), and charge order (CO) in situations where the electron-electron and electron-phonon interactions are strong (comparable to the electronic bandwidth). The model is solved in the dynamical mean-field approximation with a continuous-time quantum Monte Carlo impurity solver. We determine the superconducting transition temperature ${T}_{c}$ and the SC order parameter and show that the phonon-induced retardation or the strong Coulomb interaction leads to a significant reduction and shift of the ${T}_{c}$ dome against the effective electron-electron interaction ${U}_{\text{eff}}$ given by the Hubbard $U$ reduced by the phonon-mediated attraction in the static limit. This behavior is analyzed by comparison to an effective static model in the polaron representation with a renormalized bandwidth. In addition, we discuss the superconducting gap $\ensuremath{\Delta}$ and $2\ensuremath{\Delta}/{T}_{c}$ to reveal the effect of the retardation and the Coulomb interaction. We also determine the finite-temperature phase diagram including AF and CO. In the moderate-coupling regime, there is a hysteretic region of AF and CO around ${U}_{\mathrm{eff}}=0$, while the two phases are separated by a paramagnetic metal in the weak-coupling regime and a paramagnetic insulator in the strong-coupling regime.

arXiv:1305.5771•Physical Review B•Sep 16, 2013•62 citations
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.
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.
We study the contribution of the Higgs amplitude mode on the nonlinear optical response of superconductors beyond the BCS approximation by taking into account the retardation effect in the phonon-mediated attractive interaction. To evaluate the vertex correction in nonlinear optical susceptibilities that contains the effect of collective modes, we propose an efficient scheme which we call the ``dotted DMFT'' based on the nonequilibrium dynamical mean-field theory (nonequilibrium DMFT), to get around the difficulty of solving the Bethe-Salpeter equation and analytical continuation. The vertex correction is represented by the derivative of the self-energy with respect to the external driving field, which is self-consistently determined by the differentiated (``dotted'') DMFT equations. We apply the method to the Holstein model, a prototypical electron-phonon-coupled system, to calculate the susceptibility for the third-harmonic generation including the vertex correction. The results show that, in sharp contrast to the BCS theory, the Higgs mode can contribute to the third-harmonic generation for general polarization of the laser field with an order of magnitude comparable to the contribution from the pair breaking or charge density fluctuations. The physical origin is traced back to the nonlinear resonant light--Higgs coupling, which has been absent in the BCS approximation.
Motivated by the recent development of terahertz pump-probe experiments, we investigate the short-time dynamics in superconductors with multiple attractive pairing channels. Studying a single-band square lattice model with a spin-spin interaction as an example, we find the signatures of collective excitations of the pairing symmetries (known as Bardasis-Schrieffer modes) as well as the order parameter amplitude (Higgs mode) in the short-time dynamics of the spectral gap and quasiparticle distribution after an excitation by a pump pulse. We show that the polarization and intensity of the pulse can be used to control the symmetry of the nonequilibrium state as well as frequencies and relative intensities of the contributions of different collective modes. We find particularly strong signatures of the Bardasis-Schrieffer mode in the dynamics of the quasiparticle distribution function. Our work shows the potential of modern ultrafast experiments to address the collective excitations in unconventional superconductors and highlights the importance of subdominant interactions for the nonequilibrium dynamics in these systems.
We investigate the emergence of the collective mode in the phonon spectra of the superconducting state within the Holstein model by varying the electron-phonon coupling. Using dynamical mean field theory combined with the numerical renormalization group technique, we calculate the phonon spectra. In the superconducting state with a pairing gap ( Δ P ), the peak position of the collective mode ( ω col ) evolves from the Bardeen-Cooper-Schrieffer (BCS) regime, manifesting near 2 Δ P and increasing with coupling, to the Bose-Einstein condensation (BEC) regime, where ω col decreases with increasing coupling. The decrease of ω col matches well with the reduction of superfluid stiffness, which originates from the increasing phase fluctuations of local pairs with coupling strength. In the crossover regime with intermediate coupling, ω col aligns with the soft phonon mode ( ω s ) of the normal state and decreases with increasing coupling when ω s < 2 Δ P . Additionally, comparing the collective mode weight to Δ P suggests that the collective mode predominantly stems from U(1) gauge symmetry breaking across all coupling strengths. Published by the American Physical Society 2024
We investigate the dynamics of a phonon-mediated superconductor driven out of equilibrium. The electronic hopping amplitude is ramped down in time, resulting in an increased electronic density of states. The dynamics of the coupled electron-phonon model is investigated by solving Migdal-Eliashberg equations for the double-time Keldysh Green's functions. The increase of the density of states near the Fermi level leads to an enhancement of superconductivity when the system thermalizes to the new state at the same temperature. We provide a time- and momentum-resolved view on this thermalization process and show that it involves fast processes associated with single-particle scattering and much slower dynamics associated with the superconducting order parameter. The importance of electron-phonon coupling for the rapid enhancement and the efficient thermalization of superconductivity is demonstrated, and the results are compared to a BCS time-dependent mean-field approximation.
We study the Holstein-Hubbard model at half filling to explore ordered phases including superconductivity (SC), antiferromagnetism (AF), and charge order (CO) in situations where the electron-electron and electron-phonon interactions are strong (comparable to the electronic bandwidth). The model is solved in the dynamical mean-field approximation with a continuous-time quantum Monte Carlo impurity solver. We determine the superconducting transition temperature Tc{T}_{c}Tc​ and the SC order parameter and show that the phonon-induced retardation or the strong Coulomb interaction leads to a significant reduction and shift of the Tc{T}_{c}Tc​ dome against the effective electron-electron interaction Ueff{U}_{\text{eff}}Ueff​ given by the Hubbard UUU reduced by the phonon-mediated attraction in the static limit. This behavior is analyzed by comparison to an effective static model in the polaron representation with a renormalized bandwidth. In addition, we discuss the superconducting gap $\ensuremath{\Delta}$ and $2\ensuremath{\Delta}/{T}_{c}$ to reveal the effect of the retardation and the Coulomb interaction. We also determine the finite-temperature phase diagram including AF and CO. In the moderate-coupling regime, there is a hysteretic region of AF and CO around Ueff=0{U}_{\mathrm{eff}}=0Ueff​=0, while the two phases are separated by a paramagnetic metal in the weak-coupling regime and a paramagnetic insulator in the strong-coupling regime.