We study the dynamics of the longitudinal collective mode in an unconventional superconductor. For concreteness, we assume that the superconductor is described by a d-wave order parameter with dx2-y2 symmetry. After the superconductor has been suddenly subjected to a perturbation at time t = 0, the order parameter exhibits a peculiar oscillatory behavior, with the amplitude of the oscillations slowly decaying with time in a power-law fashion. Assuming that the initial perturbation is weak, we use a formalism based on the quasiclassical approach to superconductivity to determine both the frequency of the oscillations as well as how fast these oscillations decay with time by evaluating the time dependence of the pairing susceptibility. We find that the frequency of the oscillations is given by twice the value of the pairing amplitude in the antinodal direction and its amplitude decays as 1/t2. The results are also verified by a direct calculation of the order parameter dynamics by numerically solving the equations of motion for the Anderson pseudospins.

We study the effect of superconducting fluctuations on the spin susceptibility and NMR relaxation rate in noncentrosymmetric two-dimensional materials above the superconducting transition temperature, considering arbitrary strength of impurity scattering. Employing a microscopic model with linear Rashba spin-orbit coupling, we show that superconducting fluctuations give rise to a direct contribution to the spin susceptibility through fluctuation-induced Cooper pairs. This fluctuation-driven magnetoelectric effect is possible even for purely s-wave singlet pairing, a mechanism that is forbidden in centrosymmetric systems. It competes with the reduction of the susceptibility below the Pauli value arising from the combined effects of the suppression of the density of states and quantum-interference localization processes. In contrast, superconducting fluctuations enhance the NMR relaxation rate above its normal-state Korringa value, with spin-orbit coupling providing an additional amplification of this effect.

In superconductors with competing pairing channels, two well defined excitations exist below the pair-breaking edge: the Higgs mode of the condensed s-wave channel and the Bardasis–Schrieffer (BS) exciton of the subdominant d-wave channel. Their mixing is doubly forbidden — by point-group symmetry at zero momentum and because the two reside in the amplitude and phase sectors of the order parameter respectively, by particle–hole symmetry at every momentum. Working in a Nambu–Keldysh quasiclassical framework extended to leading 1/ε_F corrections and including the self-consistently screened Coulomb potential, we show that finite momentum combined with particle–hole asymmetry generates a direct coupling which we obtain in closed form. Whether this coupling produces an avoided crossing is decided, however, not by its magnitude but by kinematics. In the clean limit the Higgs is not a sub-gap pole but a resonance pinned to the pair-breaking edge, which disperses with coefficient unity in (v_Fq)^2, while the bound BS mode disperses more slowly: the two branches therefore separate rather than converge and never become degenerate. The obstruction is specific to the clean limit: exact dirty-limit results show that disorder detaches the amplitude resonance from the edge and reverses its dispersion, which can result in an avoided crossing with the BS mode at intermediate scattering. In that regime, the coupling computed here would set the splitting between the hybridized branches. We discuss the experimental implications of these results.

We develop a microscopic theory of the inverse Faraday effect in d-wave superconductors. An extended version of the Keldysh-Nambu quasiclassical formalism is used to compute the dc-component of the current density induced by an external monochromatic radiation. Our work explicitly demonstrates how branch population imbalance produces nonvanishing nonlinear and nonlocal dc-response. We evaluate the magnitude of the induced current and obtain estimates for the induced static magnetization. Experimental implications of our theory and future extensions of our work are briefly discussed.

We analyze the dispersion of collective modes in a superconductor with d-wave symmetry of the order parameter in the presence of long-range Coulomb interaction. We use diagrammatic technique and quasiclassical theory in the Keldysh-Nambu formalism to compute longitudinal and transverse pair susceptibilities and extract from them the dispersion of the longitudinal and transverse collective mode. We show that at T = 0 the dispersion of the transverse (plasma) mode is the same as in an s-wave superconductor, but at a finite T it is softer and has a much larger decay rate due to the partial screening of the Coulomb potential by nodal quasiparticles. We show that the dispersion of the longitudinal mode depends on the direction of momentum with respect to the positions of the nodes of the d-wave gap, while the decay rate of this mode does not depend on momentum. We discuss experimental implications of our results.

We use a self-consistent Keldysh–Nambu quasiclassical theory to study two related nonlinear phenomena in clean d-wave superconductors: the photo-induced static correction to the order parameter – the Eliashberg effect – and third-harmonic generation. Both follow from a systematic perturbative solution of the out-of-equilibrium Eilenberger equation for the Keldysh propagator. For the steady-state correction to the pairing amplitude we find that at temperatures close to the critical temperature and to leading order in the gap magnitude Δ, the photo-induced change of the order parameter is zero at all drive frequencies: in contrast to s-wave superconductors, a clean d-wave superconductor exhibits no Eliashberg enhancement at this order. The gap-enhancing quasiparticle-redistribution channel that drives the effect in the s-wave case is suppressed by an additional power of Δ in the d-wave case. For third-harmonic generation we find that the charge-density-fluctuation (particle–hole) channel significantly dominates the Schmid–Higgs amplitude-mode contribution over a broad frequency range, the two becoming comparable only in a narrow window near the resonance frequency ω≈ 2√(2) Δ if one neglects the diamagnetic part of the current in the normal state. We trace this to the nonequilibrium dynamics of nodal quasiparticles, which must be retained explicitly and which also makes the response sensitive to the orientation of the driving field.

We study spin-triplet superconductivity with both unitary and nonunitary pairing in the presence of an external Zeeman magnetic field. Within a mean-field framework, we exactly diagonalize the Bogoliubov-de Gennes Hamiltonian and derive general expressions for the quasiparticle spectrum, superconducting gap, critical temperature, and spin magnetization, valid for arbitrary magnetic-field strengths and temperatures. We analyze in detail the nonlinear spin susceptibility and the field evolution of the superconducting gap and transition temperature, highlighting qualitative differences between unitary and nonunitary pairing states. Our results are broadly applicable to a wide range of materials, including systems with both weak and strong spin-orbit coupling. We show that systematic measurements of the critical temperature and spin susceptibility as functions of the magnitude and orientation of the magnetic field provide a powerful means to identify the structure of the spin-triplet order parameter, and we discuss implications of our findings for candidate materials such as 4Hb-TaS$_2$ and PrOs$_4$Sb$_{12}$.

We revisit certain aspects of a problem concerning the influence of carrier scattering induced by magnetic impurities in metals on their superconducting properties. Superconductivity is assumed to be driven by strong electron-phonon interaction. We use the self-consistent solution of the Nagaoka equations for the scattering matrix together with the Migdal-Eliashberg theory of superconductivity to compute the energy of the in-gap bound states, superconducting critical temperature and tunneling density of states for a wide range of values of the Kondo temperature and impurity concentrations. It is found that similar to the case of the weak coupling Bardeen-Cooper-Schrieffer superconductors there is only one pair of the bound states inside the gap as well as re-entrant superconductivity for the case of antiferromagnetic exchange coupling between the conduction electrons and magnetic impurities. In addition, we find that strong electron-phonon coupling significantly reduces the decay length of the impurity-induced bound state, leading to enhanced localization compared to the weak-coupling case, which in turn narrows the spatial overlap between impurities. In agreement with the earlier studies, we find that the gapless superconductivity can be realized, which in the case of antiferromagnetic exchange requires much smaller impurity concentration. Surprisingly, in contrast with the weakly coupled superconductors we find that superconducting transition exhibits two critical temperatures for the ferromagnetic exchange coupling.

We compute the second-harmonic response of two-dimensional topological Dirac semimetals subjected to an external in-plane magnetic field. The quantum kinetic equation for the Wigner distribution function is derived and then solved to evaluate the second-order electric-field contributions to the current density. Both the Berry curvature dipole and the field-induced terms in the current are analyzed across a broad range of model parameters. We propose that our theory can be tested experimentally by measuring the dependence of the anomalous Hall resistivity on the in-plane magnetic field in the surface states of the topological insulator SnTe, in WTe_2 and WSe_2 monolayers, as well as in the Kondo lattice material Ce_3Bi_4Pd_3 at very low temperatures.

We consider the nonlinear response of a disordered two-dimensional electronic system, lacking inversion symmetry, to an external alternating electric field. The application of an in-plane static magnetic field induces local contributions to the current density that are quadratic in the electric field and linear in the magnetic field. This current oscillates at twice the frequency of the external irradiation and there are two linearly independent vector combinations that contribute to the current density. This particular mechanism coexists with the topological Berry-dipole contribution to the second harmonic of the current density, which can be generated by quantum confinement. Additional nonlocal terms in the current density are possible in the regime away from the normal incidence. The total current exhibits a nonreciprocal character upon reversal of the magnetic field direction. We evaluate the magnitude of this effect by computing its dependence on the strength of spin-orbit coupling and the disorder scattering rate. Importantly, we show that these local second-harmonic contributions can be resonantly excited when the frequency of the external radiation approaches the energy separation between the spin-orbit split bands.

Motivated by the recent experimental measurements of the nonlinear longitudinal resistance of the spin-orbit coupled electron gas in the (111) LaTiO3/SrTiO3 interfaces under external in-plane magnetic field [G. Tuvia, et al., Phys. Rev. Lett. 132, 146301 (2024)], we formulate a theory of nonlinear electronic transport based on the analysis of the quantum kinetic equation for the Wigner distribution function. Specifically, we evaluate the magnetic field dependence of the second harmonic of the current density at arbitrary values of the magnetic field. The magnitude of the second harmonic increases linearly with the magnetic field at small fields. Upon further increase of the magnetic field, the second harmonic response reaches its maximum value. We find that the position of the peak and its width strongly depend on the relaxation rate due to disorder. Importantly, we discover that the direction of the nonlinear contribution to the current can be completely reversed when the magnetic field reaches a certain critical value.

We consider a problem of nonlinear response to an external electromagnetic radiation in conventional disordered superconductors which contain a small amount of weak magnetic impurities. We focus on the diffusive limit and use Usadel equation to analyze the excitation energy and dispersion relation of the collective modes. We determine the resonant frequency and dispersion of both amplitude (Schmidt-Higgs) and phase (Carlson-Goldman) modes for moderate strength of magnetic scattering. We find that the minimum energy required for the excitation of the both of these modes decreases with an increase in spin-flip scattering. Surprisingly we also find that as a result the Carlson-Goldman mode becomes gapless and as a consequence can only be excited at some finite value of the threshold momentum. We thus discover yet another physical realization of a state with gapped momentum dispersion of one of its collective modes. The value of the threshold momentum is determined by the distance between the two consecutive spin-flip scattering events which, in turn, is proportional to the scattering time between two consecutive scattering events. The amplitude mode is diffusive and becomes strongly suppressed with the increase in spin-flip scattering. Possible ways to experimentally verify our results are also discussed.

I formulate a theory of the inverse Faraday effect in an impure two-dimensional metallic system with lifted spin degeneracy induced by Rashba spin-orbit coupling. Using the formalism of nonequilibrium quantum field theory, the static contributions to the current density up to the second order in powers of the external electromagnetic field are evaluated. For circularly polarized light one of the contributions describes the emergence of static magnetization. It is shown that the direction of induced magnetization may change depending on the frequency of the external radiation and disorder scattering rate. I also find that at large frequencies the leading contribution to the induced magnetization is proportional to the square of the spin-orbit interaction and is inversely proportional to the fifth power of the frequency of an external electromagnetic wave.

Collective modes in superconductors provided the first realization of the Higgs mechanism. The transverse Goldstone mode acquires a gap (i.e. a mass) when it hybridizes with the electromagnetic gauge field. The longitudinal Schmid-Higgs mode, on the other hand, is always massive. In conventional BCS theory, its gap is exactly 2 Delta, coinciding with the excitation threshold for quasiparticles. Being situated right at the edge of the continuum spectrum it gives rise to peculiar dynamics for the Schmid-Higgs mode. For instance, when suddenly excited at t=0, it exhibits algebraically decaying oscillations of the form similar to sin(2 Delta t)/t(1/2). In this study, we explore the behavior of Schmid-Higgs oscillations in the presence of pair-breaking mechanisms, such as magnetic impurities or in-plane magnetic fields. These processes suppress the quasiparticle excitation threshold down to 2 epsilon(g)<2 Delta, potentially placing the longitudinal mode within the continuum spectrum. Despite this, we show that the algebraically decaying oscillations persist, taking the form similar to sin(2 epsilon(g)t)/t(2). The Schmid-Higgs mode becomes truly overdamped and exponentially decaying only in the gapless superconductors with epsilon(g)=0.

Motivated by the recent experimental measurements of the nonlinear longitudinal resistance of the spin-orbit coupled electron gas in the (111) LaTiO_3/SrTiO_3 interfaces under external in-plane magnetic field [G. Tuvia et al., Phys. Rev. Lett. 132, 146301 (2024)], we formulate a theory of nonlinear electronic transport based on the analysis of the quantum kinetic equation for the Wigner distribution function. Specifically, we evaluate the magnetic field dependence of the second harmonic of the current density at arbitrary values of the magnetic field. The magnitude of the second harmonic increases linearly with the magnetic field at small fields. Upon further increase of the magnetic field, the second harmonic response reaches its maximum value. We find that the position of the peak and its width strongly depend on the relaxation rate due to disorder. Importantly, we discover that the direction of the nonlinear contribution to the current can be completely reversed when the magnetic field reaches a certain critical value.

We study nonlinear response of conventional superconducting alloys with weak magnetic impurities to an external alternating electromagnetic field. In particular, we calculate a correction to superconducting order parameter up to the second order in external vector potential. We show that frequency dependence of the order parameter amplitude has characteristic resonant shape with a maximum at frequency which is smaller than twice the magnitude of the pairing amplitude in equilibrium and at the same time exceeds the single-particle threshold energy. Our results suggest that in the presence of magnetic impurities the dynamics of the pairing amplitude in the collisionless regime will remain robust with respect to dissipative processes. We also evaluate the third harmonic contribution to the current as a function of the probe frequency and for various concentrations of magnetic impurities.

We apply the Migdal-Eliashberg theory of superconductivity to heavy-fermion and mixed valence materials. Specifically, we extend the Anderson lattice model to a case when there exists a strong coupling between itinerant electrons and lattice vibrations. Using the saddle-point approximation, we derive a set of coupled nonlinear equations which describe competition between the crossover to a heavy-fermion or mixed-valence regimes and conventional superconductivity. We find that superconductivity at strong coupling emerges on par with the development of the many-body coherence in a Kondo lattice. Superconductivity is gradually suppressed with the onset of the Kondo screening and for strong electron-phonon coupling the Kondo screening exhibits a characteristic re-entrant behavior. Even though for both weak and strong coupling limits the suppression of superconductivity is weaker in the mixed-valence regime compared to the local moment one, superconducting critical temperature still remains nonzero. In the weak coupling limit the onset of the many body coherence develops gradually, in the strong coupling limit it emerges abruptly in the mixed valence regime while in the local moment regime the $f$-electrons remain effectively decoupled from the conduction electrons. Possibility of experimental realization of these effects in Ce-based compounds is also discussed.

I consider a nonlinear response of conventional superconductors contaminated with potential impurities or imperfections to a circular polarized light. I focus on dc contributions to the induced current density which describe the emergence of the static magnetization in a superconductor. This effect is known as inverse Faraday effect. By employing quasiclassical theory of superconductivity I derive an expression for the induced static magnetization as a function of frequency of an external ac field and disorder scattering rate. The scattering of electrons off potential impurities is taken into account within the framework of the self-consistent Born approximation. It is found that the magnitude of the inverse Faraday effect decreases with an increase in disorder scattering rate. I have also discovered that the value of the induced magnetization has a characteristic minimum at a frequency which approximately equals twice the value of the pairing gap in a clean superconductor and shifts to higher values with an increase in disorder scattering rate.