Third-harmonic generation (THG) experiments on superconductors can be used to investigate collective excitations like the amplitude mode of the order parameter known as Higgs mode. These modes are visible due to resonances in the THG signal if the driving frequency matches the energy of the mode. In real materials multiple modes can exist giving rise to additional THG contributions, such that it is difficult to unambiguously interpret the results. In this paper, we additionally analyze the phase of the THG signal, which contains microscopic details beyond classical resonances as well as signatures of couplings between modes which are difficult to observe in the amplitude alone. We investigate how the Higgs mode, impurities or Coulomb interaction affects the phase response and consider exemplary two systems with additional modes. We argue that extracting this phase information could be valuable in future experiments.
We investigate the behavior of the Higgs (amplitude) mode when superconductivity emerges on a preexisting charge-density-wave state. We show that the weak overdamped square-root singularity of the amplitude fluctuations in a standard BCS superconductor is converted in a sharp, undamped power-law divergence in the coexisting state, reminiscent of the Higgs behavior in Lorentz-invariant theories. This effect reflects in a strong superconducting resonance in the Raman spectra, both for an electronic and a phononic mechanism leading to the Raman visibility of the Higgs. In the latter case, our res…
Low temperature thermal conductivity in a d-wave superconductor with coexisting charge order: Effect of self-consistent disorder and vertex corrections
We study thermal transport in a two-dimensional system with coexisting $s$- or $d$-wave Superconducting (SC) and Spin Density Wave (SDW) orders. We analyse the nature of coexistence phase in a tight-binding square lattice with $\mathbf{Q}=(π,π)$ SDW ordering. The electronic thermal conductivity is computed within the framework of the Boltzmann kinetic theory, using Born approximation for the impurity scattering collision integral. We describe the influence of the Fermi surface (FS) topology, the competition between the SC and SDW order parameters, the presence or absence of zero energy excitat…
Higgs spectroscopy is a new field in which Higgs modes in nonequilibrium superconductors are analyzed to gain information about the ground state. One experimental setup in which the Higgs mode in s-wave superconductors was observed is periodic driving with THz light, which shows resonances in the third-harmonic generation (THG) signal if twice the driving frequency matches the energy of the Higgs mode. We derive expressions of the driven gap oscillations for arbitrary gap symmetry and calculate the THG response. We demonstrate that the possible Higgs modes for superconductors with non-trivial …
In superconductors the Anderson-Higgs mechanism allows for the existence of a collective amplitude (Higgs) mode which can couple to eV-light mainly in a non-linear Raman-like process. The experimental non-equilibrium results on isotropic superconductors have been explained going beyond the BCS theory including the Higgs mode. Furthermore, in anisotropic d-wave superconductors strong interaction effects with other modes are expected. Here we calculate the Raman contribution of the Higgs mode from a new perspective, including many-body Higgs oscillations effects and their consequences in convent…
Anderson-Higgs mechanism of mass generation is a generic concept in high-energy and condensed matter physics. It shows up through the Meissner effect providing the expulsion of static and low-frequency magnetic fields from superconductors. However, it does not affect propagating electromagnetic waves with a spectrum gap determined by the plasma frequency, which is too large to be sensitive to the superconducting transition. Here we demonstrate the spectroscopic manifestation of the Anderson-Higgs mass, showing that it determines the spectrum gap of magnons in superconductor/ferromagnet/superco…
The details of the superconducting to quantum metal transition (SQMT) at T=0 are an open problem that invokes much interest in the nature of this exotic and unexpected ground state1-3. However, the SQMT was not yet investigated in a crystalline 2D superconductor with coexisting and fluctuating quantum orders. Here, we report the observation of a SQMT in 2D ion-gel gated 1T-TiSe24, driven by magnetic field. A field-induced crossover between Bose quantum metal and vortex quantum creeping with increasing field is observed. We discuss the interplay between superconducting and CDW fluctuations (dis…
The concept of mass-generation via the Higgs mechanism was strongly inspired by earlier works on the Meissner-Ochsenfeld effect in superconductors. In quantum field theory, the excitations of longitudinal components of the Higgs field manifest as massive Higgs bosons. The analogous Higgs mode in superconductors has not yet been observed due to its rapid decay into particle-hole pairs. Following recent theories, however, the Higgs mode should decrease below the pairing gap $2Δ$ and become visible in two-dimensional systems close to the superconductor-insulator transition (SIT). For experimental…
We investigate theoretically the excitation of Higgs oscillations of the order parameter in superconductors by incoherent short pulses of light with frequency much larger than the superconducting gap. The excitation amplitude of the Higgs mode is controlled by a single parameter which is determined by the total number of quasiparticles excited by the pulse. This fact can be traced back to the universality of the shape of the light-induced quasiparticle cascade at energy below the Debye frequency and above the gap.
Higgs mode in superconducting materials describes slowly-decaying oscillations of the order parameter amplitude. We demonstrate that in magnetic superconductors with built-in spin-splitting field Higgs mode is strongly coupled to the spin degrees of freedom allowing for the generation of time-dependent spin currents. Converting such spin currents to electric signals by spin-filtering elements provides a tool for the second-harmonic generation and the electrical detection of the Higgs mode generated by the external irradiation. The non-adiabatic spin torques generated by these spin currents all…
We reveal that due to the presence of disorder oscillations of the order parameter amplitude called the Higgs mode can be effectively excited by the external electromagnetic radiation in usual BCS superconductors. This mechanism works for superconductors with both isotropic s-wave and anisotropic, such as d-wave, pairings. The non-linear response in the presence of impurities is captured by the quasiclassical formalism. We demonstrate that analytical solutions of the Eilenberger equation with impurity collision integral and external field drive coincide with the exact summation of ladder impur…
Collective excitation modes are a characteristic feature of symmetry-broken phases of matter. For example, superconductors exhibit an amplitude Higgs mode and a phase mode, which are the radial and angular excitations in the Mexican-hat potential of the free energy. In two-band superconductors there exists in addition a Leggett phase mode, which corresponds to collective fluctuations of the interband phase difference. In equilibrium systems amplitude and phase modes are decoupled, since they are mutually orthogonal excitations. The direct detection of Higgs and Leggett modes by linear-response…
We put forth a mechanism for enhancing the interlayer transport in cuprate superconductors, by optically driving plasmonic excitations along the $c$ axis with a frequency that is blue-detuned from the Higgs frequency. The plasmonic excitations induce a collective oscillation of the Higgs field which induces a parametric enhancement of the superconducting response, as we demonstrate with a minimal analytical model. Furthermore, we perform simulations of a particle-hole symmetric $U(1)$ lattice gauge theory and find good agreement with our analytical prediction. We map out the renormalization of…
Through the Higgs mechanism, the long-range Coulomb interaction eliminates the low-energy Goldstone phase mode in superconductors and transfers spectral weight all the way up to the plasma frequency. Here we show that the Higgs mechanism breaks down for length scales shorter than the superconducting coherence length while it stays intact, even at high energies, in the long-wavelength limit. This effect is a consequence of the composite nature of the Higgs field of superconductivity and the broken Lorentz invariance in a solid. Most importantly, the breakdown of the Higgs mechanism inside the s…
Quantum fluctuations and the emergence of in-gap Higgs mode in superconductors
Max Planck Institute for Solid State Research（马克斯·普朗克固体研究所）；The University of Tokyo（东京大学）；RIKEN Center for Emergent Matter Science (CEMS)（理化学研究所凝聚体物质科学中心）；Trans-Scale Quantum Science Institute, The University of Tokyo（东京大学跨尺度量子科学研究院）
In a recent manuscript Phys. Rev. Lett. 127, 197001 (2021) Puviani et al. claim that the Higgs mode gives an enhancement of the Raman response in the symmetric $A_{1g}$ Raman channel of cuprates. Here we report a series of technical mistakes in their derivation that completely invalidate this conclusion. In particular, the Raman response function contains two renormalized vertices, with a clearly overcounting of diagrams and uncontrolled results for the Higgs-mode contribution.
Type II Lifshitz invariant and optically active Higgs mode in time-reversal symmetry broken superconductors
Institute for Theory of Condensed Matter, Karlsruhe Institute of Technology（卡尔斯鲁厄理工学院凝聚态理论研究所）；Department of Physics, University of Tokyo（东京大学物理学系）；RIKEN Center for Emergent Matter Science (CEMS)（理化学研究所新兴物质科学中心）；Trans-Scale Quantum Science Institute, University of Tokyo（东京大学跨尺度量子科学研究所）
The recent observation of a transmitted Thz pulse oscillating at three times the frequency of the incident light paves the way to a new protocol to access resonant excitations in a superconductor. Here we show that this non-linear optical process is dominated by light-induced excitation of Cooper pairs, in analogy with a standard Raman experiment. The collective amplitude (Higgs) fluctuations of the superconducting order parameter give in general a smaller contribution, unless one designs the experiment by combining properly the light polarization with the lattice symmetry.
Topological superconductors have attracted tremendous excitement as they are predicted to host Majorana zero modes that can be utilized for topological quantum computing. Candidate topological superconductor Sn1-xInxTe thin films (0<x<0.3) grown by molecular beam epitaxy and strained in the (111) plane are shown to host three coexisting quantum effects: localization, antilocalization and superconducting fluctuations above the critical temperature Tc. An analysis of the normal state magnetoresistance reveals these effects. Weak localization is consistently observed in superconducting samples, i…
Recent studies have emphasized the importance of impurity scattering for the optical Higgs response of superconductors. In the dirty limit, an additional paramagnetic coupling of light to the superconducting condensate arises which drastically enhances excitation. So far, most work concentrated on the periodic driving with light, where the third-harmonic generation response of the Higgs mode was shown to be enhanced. In this work, we additionally calculate the time-resolved optical conductivity of single- and two-band superconductors in a two-pulse quench-probe setup, where we find good agreem…
We propose to induce a time crystalline state in a high-$T_c$ superconductor, by optically driving a sum resonance of the Higgs mode and a Josephson plasma mode. The generic cubic process that couples these fundamental excitations converts driving of the sum resonance into simultaneous resonant driving of both modes, resulting in an incommensurate subharmonic motion. We use a numerical implementation of a semi-classical driven-dissipative lattice gauge theory on a three-dimensional layered lattice, which models the geometry of cuprate superconductors, to demonstrate the robustness of this moti…
Despite the formal analogy with the Higgs particle, the amplitude fluctuations of the order parameter in weakly-coupled superconductors do not identify a real mode with a Lorentz-invariant dynamics. Indeed, its resonance occurs at $2Δ_0$, which coincides with the threshold $2E_{gap}$ for quasiparticle excitations, that spoil any relativistic dynamics. Here we investigate the fate of the Higgs mode in the unconventional case where $2E_{gap}$ becomes larger than $2Δ_0$, as due to strong coupling or strong disorder. We show that also in this situation the amplitude fluctuations never identify a r…
Using the non-equilibrium Keldysh formalism, we solve the equations of motion for electron-phonon superconductivity, including an ultrafast pump field. We present results for time-dependent photoemission spectra out of equilibrium which probes the dynamics of the superconducting gap edge. The partial melting of the order by the pump field leads to oscillations at twice the melted gap frequency, a hallmark of the Higgs or amplitude mode. Thus the Higgs mode can be directly excited through the nonlinear effects of an electromagnetic field and detected without any additional symmetry breaking.
Higgs oscillations in nonequilibrium superconductors provide an unique tool to obtain information about the underlying order parameter. Several properties like the absolute value, existence of multiple gaps and the symmetry of the order parameter can be encoded in the Higgs oscillation spectrum. Studying Higgs oscillations with time-resolved angle-resolved photoemission spectroscopy (ARPES) has the advantage over optical measurements that a momentum-resolved analysis of the condensate dynamic is possible. In this paper, we investigate the time-resolved spectral function measured in ARPES for d…
The pressure evolution of the Raman active electronic excitations of the transition metal dichalcogenides 2H-TaS$_2$ is followed through the pressure phase diagram embedding incommensurate charge-density-wave and superconducting states. At high pressure, the charge-density-wave is found to collapse at 8.5~GPa. In the coexisting charge-density-wave and superconducting orders, we unravel a strong in-gap superconducting mode, attributed to a Higgs mode, coexisting with the expected incoherent Cooper-pair breaking signature. The latter remains in the pure superconducting state reached above 8.5~GP…
We develop a microscopic and gauge-invariant theory for collective modes resulting from the phase of the superconducting order parameter in non-centrosymmetric superconductors. Considering various crystal symmetries we derive the corresponding gauge mode $ω_{\rm G}({\bf q})$ and find, in particular, new Leggett modes $ω_{\rm L}({\bf q})$ with characteristic properties that are unique to non-centrosymmetric superconductors. We calculate their mass and dispersion that reflect the underlying spin-orbit coupling and thus the balance between triplet and singlet superconductivity occurring simultane…
We consider the amplitude (Higgs) mode in a superconductor with a condensate flow (supercurrent). We demonstrate that, in this case, the amplitude mode corresponding to oscillations $δ|Δ|_Ω \exp(i Ωt)$ of the superconducting gap is excited by an external ac electric field $\mathbf{E}_Ω \exp(i Ωt)$ already in the first order in $|\mathbf{E}_Ω|$, so that ${δ|Δ|_Ω \propto (\mathbf{v}_{0} \mathbf{E}_Ω)}$, where $\mathbf{v}_{0}$ is the velocity of the condensate. The frequency dependence $δ|Δ|_Ω$ has a resonance shape with a maximum at ${Ω= 2 Δ}$. In contrast to the standard situation without the c…
