Optical response of Higgs mode in superconductors at clean limit

Yang, Fang; Wu, M. W.

doi:10.48550/arxiv.2111.06128

Cited by 2 publications

(10 citation statements)

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“…41, the involved optical frequency Ω is taken as bosonic Matsubara frequency iΩ m , leading to a vanishing response coefficient λ E in Eq. ( 1) strongly against the finite one from gauge-invariant kinetic equation 19,46 and Ginzburg-Landau Lagrangian 46 . Moreover, because of this treatment, the prefactor 1/Ω 2 in Eq.…”

Section: Introductionmentioning

confidence: 98%

“…This finite Higgs-mode generation has therefore been in sharp contrast to the aforementioned vanishing one from Eilenberger equation and path-integral approach. Actually, at clean limit, the finite Higgs-mode generation in second-order response of superconductors is a direct consequence of the Ginzburg-Landau Lagrangian 46 . This is because that from the time-dependent Ginzburg-Landau superconducting Lagrangian at clean limit, which was proposed by Pekker and Varma through the symmetry analysis and Lorentz invariance from the Landau phasetransition theory 6 , one can directly reveal the equation of motion of the Higgs mode by considering the amplitude fluctuation of the Landau order parameter.…”

Section: Introductionmentioning

confidence: 99%

“…This is because that from the time-dependent Ginzburg-Landau superconducting Lagrangian at clean limit, which was proposed by Pekker and Varma through the symmetry analysis and Lorentz invariance from the Landau phasetransition theory 6 , one can directly reveal the equation of motion of the Higgs mode by considering the amplitude fluctuation of the Landau order parameter. Then, a finite Higgs-mode generation in the second-order response is immediately obtained 46 . This directly leads to a particularly bizarre question that why both path-integral approach 47 and Eilenberger equation 48 can recover the Ginzburg-Landau equation but reach a zero Higgs-mode generation.…”

Section: Introductionmentioning

confidence: 99%

“…41, it is pointed out by Yang and Wu 46 that both previous derivations contain flaws. Specifically, the previous works [34][35][36] within the path-integral approach only kept the per-turbation expansion of the action up to second order, whereas the essential coupling of the Higgs mode to the second order of the drive effect H d emerges in the thirdorder perturbation expansion of the action 46 . So this coupling and hence finite second-order harmonic generation of Higgs mode by the drive effect are excessively overlooked in Refs.…”

Section: Introductionmentioning

confidence: 99%

“…Moreover, because of this treatment, the prefactor 1/Ω 2 in Eq. ( 1) causes an undefined singularity at zero frequency, and an unphysical discontinuity between cases at Ω = 0 and Ω → 0 emerges 46 . In contrast, the previous work in Ref.…”

Section: Introductionmentioning

confidence: 99%

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Optical response of Higgs mode in superconductors at clean limit: formulation through Eilenberger equation and Ginzburg-Landau Lagrangian

Yang¹,

Wu²

2023

Preprint

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Both macroscopic Ginzburg-Landau Lagrangian and microscopic gauge-invariant kinetic equation suggest a finite Higgs-mode generation in the second-order optical response of superconductors at clean limit, whereas the previous derivations through the path-integral approach and Eilenberger equation within the Matsubara formalism failed to give such generation. The crucial treatment leading to this controversy lies at an artificial scheme that whether the external optical frequency is taken as continuous variable or bosonic Matsubara frequency to handle the gap dynamics within the Matsubara formalism. To resolve this issue, we derive the effective action of the superconducting gap near Tc in the presence of the vector potential through the path-integral approach, to fill the long missing blank of the microscopic derivation of the Ginzburg-Landau Lagrangian in superconductors. It is shown that only by taking optical frequency as continuous variable within the Matsubara formalism, can one achieve the fundamental Ginzburg-Landau Lagrangian, and in particular, the finite Ginzburg-Landau kinetic term leads to a finite Higgs-mode generation at clean limit. To further eliminate the confusion of the Matsubara frequency through a separate framework, we apply the Eilenberger equation within the Keldysh formalism, which is totally irrelevant to the Matsubara space. By calculating the gap dynamics in the second-order response, it is analytically proved that the involved optical frequency is a continuous variable rather than bosonic Matsubara frequency, causing a finite Higgs-mode generation at clean limit.

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Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Optical response of Higgs mode in superconductors at clean limit: formulation through Eilenberger equation and Ginzburg-Landau Lagrangian

Yang¹,

Wu²

2023

Preprint

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Recent Development of Ultrafast Optical Characterizations for Quantum Materials

2022

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or far-infrared energy region are therefore of particular importance, for the THz frequencies (1 THz≈4.1 meV) are close to the energy scales of a number of important single-particle and collective excitations of quantum material systems, for example, pairing energy gaps of superconductors, lattice vibration modes (phonons), collective spin waves (magnons), Josephson plasmon in high temperature superconductors, and binding energy of bound electron-hole pairs (excitons), [1] and they are linked to a large number of fundamental physical problems currently under intensive investigations in condensed matter physics. The information yielded from equilibrium optical measurements are usually prerequisites for understanding the nonequilibrium optical properties probed by ultrafast time-resolved spectroscopy techniques.In the last three decades, we have witnessed a rapid development in ultrafast laser and nonlinear optical techniques. The intense ultrashort optical pulses at wavelengths spanning the electromagnetic spectrum from terahertz through visible to X-ray have been generated from table-top laser sources and free-electron laser facilities. A variety of time-resolved spectroscopy techniques on the basis of pump-probe scheme have been developed. Depending on the fluence of pump pulses, photoexcitations can induce different effects and phenomena. For sufficiently weak perturbations, the photoexcitation of charge carriers does not change the free-energy landscape and the system relaxes back to the equilibrium state in a short time scale. The relaxation process is determined by the same interactions (e.g for example, electron-electron, electron-phonon) that govern the equilibrium properties. In principle, one can extract the coupling strength of electron to other degrees of freedom from the relaxation dynamics. In addition, the ultrashort laser pulse whose duration is shorter than the periodicity of targeted lattice/spin modes can trigger coherent oscillations at the corresponding frequencies of collective modes, for example, phonon, magnon, etc. with decay times that are related to the dephasing times. This enables one to detect collective-mode excitations that may not be easily accessed to or probed by other techniques.By contrast, with strong pump pulses, the photoexcitation can excite a sufficient number of electrons or drive the motion of collective mode to a large amplitude. The corresponding perturbation to the lattice/spin degrees of freedom can be large enough to modify the free-energy landscape. As a result, The advent of intense ultrashort optical pulses spanning a frequency range from terahertz to the visible has opened a new era in the experimental investigation and manipulation of quantum materials. The generation of strong optical field in an ultrashort time scale enables the steering of quantum materials nonadiabatically, inducing novel phenomenon or creating new phases which may not have an equilibrium counterpart. Ultrafast time-resolved optical techniques have provided rich information and played an im...

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Optical response of Higgs mode in superconductors at clean limit

Cited by 2 publications

References 59 publications

Optical response of Higgs mode in superconductors at clean limit: formulation through Eilenberger equation and Ginzburg-Landau Lagrangian

Optical response of Higgs mode in superconductors at clean limit: formulation through Eilenberger equation and Ginzburg-Landau Lagrangian

Recent Development of Ultrafast Optical Characterizations for Quantum Materials

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