Nonequilibrium Dynamics:from Spin Dynamics In Two-dimensional Materials To Electromagnetic Response Of Superconductors

Within the equal-time nonequilibrium Green function approach to set up the micro-scopic kinetic equation,this dissertation,starting with the kinetic spin Bloch equation of spintronics,first focuses on the spin dynamics in two-dimensional materials as the prologue.Then,in the main part—the field of superconductors,a microscopic kinetic theory named as gauge-invariant kinetic equation is developed,in order to study the rich electromagnetic properties of superconductors.In the prologue,we use the kinetic spin Bloch equation to study the hole spin dynamics in bilayer transition metal dichalcogenides,including the spin relaxation and diffusion in K and K' valleys under the influence from Rashba spin-orbit coupling.Due to the property of bilayer materials,we obtain different spin relaxation(diffusion)processes in the two valleys.Particularly,in the relaxation(diffusion)process of large spin polarization,we find that the initially equal hole densities in the two valleys are broken during the temporal evolution(along the diffusion),leading to the built-up of a nonequilibrium(steady-state)valley polarization.In the main part,focusing on the field of superconductors,we first investigate the equilibrium superconductivity in systems with the translational-symmetry breaking,and study the nonequilibrium electromagnetic response of superconductors afterwards.The investigation on the equilibrium is based on the Gorkov equation.Based on the symmetry analysis,we first discuss the requirement to realize unconventional Cooper pairs in the presence of the translational-symmetry breaking.Then,it is shown that all four types(singlet even-frequency,singlet odd-frequency,triplet even-frequency and triplet odd-frequency)of Cooper pairs can be realized in spin-orbit-coupled quantum well in proximity to s-wave superconductor with the translational-symmetry breaking.While the self-energy of the electron-electron Coulomb interaction with the inevitable plasmon effect in the two-dimensional quantum well leads to all four types of the order parameters.After that,in spin-orbit-coupled s-wave superconductors,we discuss the possibility to break the translational-symmetry breaking(induce the center-of-mass momentum of Cooper pair)by using the Zeeman effect of magnetic field.We show that the magnetic field leads to two supcrconducting phases with the center-of-mass momenetum of Cooper pair:drift.-BCS state at small nagnetic field and Fulde-Fcrrell state at largo one.In the former state.the center-of-mass momentum of Cooper pair originates from the energy-speetrum distortion,whereas the latter state is similar to the conventional Fulde-Ferrell stateActually,the Gorkov equation is very hard to handle for a kinetic calculation of the nonequilibrium properties as too many variables are involved in the Green function.Thus,the development of microscopic kinetic equation for nonequilibrium dynamics in superconductors is necessary.To start,by first using the gauge-invariant optical Bloch equation approach established by Yu and Wu,we study the anomalous Hall effect in chiral p-wave superconducting states.It is demonstrated that the intrinsic anomalous Hall conductivity is zero as a consequence of Galilean invariance,whereas the impurity scattering leads to the extrinsic one.In contrast to the Kubo formalism for describing the conventional skew-scattering channel in the linear response,our study not only provides a microscopic kinetic description of this channel,but also reveals a new induction channel from the Born contribution by nonlinear optical excitation,which dominates the anomalous Hall conductivity at weak impurity interaction.After that,we develop the gauge-invariant optical Bloch equation to include the complete superfluid dynamics and electromagnetic effects,and then,establish the gauge-invariant kinetic equation of superconductors.We first prove that the gauge-invariant kinetic equation satisfies the gauge structure revealed by Nambu,and then,the retained gauge invariance directly leads to the charge conservation in the electromagnetic response.Through the gauge-invariant kinetic equation,we study the current excitation in the mag-netic and low-frequency optical responses.Besides the recoveries of the well-known results in the literature,such as the Ginzburg-Landau equation and Meissner supercurrent in the magnetic response as well as the optical current captured by the two-fluid model,we predict that the normal fluid and hence the scattering are present only when the excited superconducting velocity is larger than a threshold.Particularly,we find that there exists friction between the normal-fluid and superfluid currents.Part of the superfluid becomes viscous due to this friction.Therefore,a three-fluid model:normal fluid,non-viscous and viscous superfluids,is proposed to capture the electromagnetic response of superconduct-ing states.Rich physical behaviors,including the origin of the scattering influence on the penetration depth,a modified Ginzburg-Landau equation,an exotic phase with both finite resistivity and superconducting gap as well as optical conductivity captured by the three-fluid model,are then revealed.Furthermore,we show that the gauge-invariant kinetic equation of superconductors provides an efficient approach to calculate the electromagnetic responses of the collective Nambu-Goldstone and Higgs modes on an equal footing.We first recover the conventional results of the linear responses of both collective modes in the literature.Then,we show that the second-order optical response of the Higgs mode is attributed solely to the drive effect(i.e.,drive effect of optical electric field and diamagnetic effect of vector potential)rather than the widely considered paramagnetic effect of vector potential in the literature.Particularly,we also find a finite second-order response of the Nambu-Goldstone mode,which decouples with the long-range Coulomb interaction as a consequence of the charge conservation and hence becomes free from the influence of the Anderson-Higgs mechanism.A tentative scheme to detect this second-order response is then proposed.In addition,we also show that the gauge-invariant kinetic equation of superconductors provides an efficient approach to calculate the scattering effect.It is found that the obtained optical absorption in the linear regime,induced by the scattering,can well capture the experimentally observed features in dirty superconductors lying in the normal-skin-effect region.While in the second-order regime,we find that the scattering causes a phase shift in the optical response of the Higgs mode,which exhibits a significant ?-junmp at??|?|.Additionally,it is also found that the elastic impurity scattering causes the damping of the Higgs-mode excitation after the pulse.In short,we have proved/shown that our gauge-invariant kinetic equation not only involves both superfluid and normal-fluid dynamics,but also is capable of calculating both magnetic and optical responses in linear and nonlinear regimes.Due to the retained gauge invariance,the charge conservation that is crucial for electromagnetic properties is natu-rally satisfied in the gauge-invariant kinetic equation.Furthermore,we have also shown that the gauge-invariant kinetic equation provides an efficient approach to calculate the electromagnetic response of the various collective modes in superconductors.Additionally,thanks to the equal-time noncquilibrium Green-function approach,the gauge-invariant ki-netic equation contains the microscopic scattering terms to elucidate the scattering effect.Through this theory,besides the recoveries of the well-known results in the literature,rich physics has been revealed.The gauge-invariant kinetic equation therefore provides an ef-ficient approach to calculate the nonequilibrium dynamics and electromagnetic properties in superconductors.One may expect much more rich and interesting physics from this theory.Finally,by applying the gauge-invariant kinetic equation approach in d-wave super-conductors,we analytically derive the energy spectra of the breathing Higgs mode and the rotating Higgs mode that is unique for the d-wave order parameter.Then,the dynamic properties of both Higgs modes,including the optical and magnetic responses as well as the negative thermal Hall signal which has attracted much interests recently,are presented.The content of this dissertation mostly focuses on the analytical investigation.To make it easier for the readers,we only presents the specific model,results and analysis in the main tcxt,wlcreas the lengthy derivations are placed in the ten appendixes.More detailed abstracts of eachc chapters are given in the following.In the prologue,from Chapter 1 to 2,we perform the investigation on the spin dy-namics in bilayer transition metal dichalcogenides.In Chapter 1,we introduce the two-dimensional monolayer and bilayer transition-metal dichalcogenides,and the progress of the valleytronics(including the valley Hall effect of free carriers,valley polarization and depolarization mechanisms of exciton)and spintronics(including the spin generation,spin detection,spin-relaxation mechanisms and spin-diffusion models)in these kinds of ma-terials.In bilayer transition-metal dichalcogenides,one can not only realize the spatial separation of different hole spins in K and K' valleys via spin-layer locking effect,but also allow the optical creation of the hole spin polarization in two valleys through the chiral optical selection rule.This kind of materials thus provides an ideal platform to explore the spin dynamics and a promising candidate for possible spintronic application.The understanding of hole spin relaxation and diffusion then becomes an important topic.Focusing on this topic,in Chapter 2,we first introduce the kinetic spin Bloch equa-tion in spintronics.The kinetic spin Bloch equation was established and developed by Wu via extending the optical Bloch equation of semiconductors into the spin space within the equal-time nonequilibrium Green function approach.This equation not only contains the microscopic scattering effect,but also handles the many-body effect pretty well.Through the kinetic spin Bloch equation,we investigate the hole spin dynamics in bilayer transition-metal dichalcogenides.In consideration of the electric manipulation of hole density in the experiments,we focus on the influence on both the spin relaxation and diffusion from the Rashba spin-orbit coupling induced by external gate voltage.In contrast to the con-ventional in-plane form,the Rashba spin-orbit coupling in bilayer materials possesses an additional out-of-plane component,which acts as opposite Zeeman-like fields in K and K'valley and leads to the rich physics of spin dynamics.For spin relaxation,this Zeeman-like field,together with the intervalley hole-phonon scattering,opens an intervalley in-plane spin relaxation channel,which dominates the in-plane spin relaxation.For out-of-plane spins,the Zeeman-like field is superimposed by the identical Hartree-Fock effective magnetic fields in the two valleys.Then,different total effective magnetic fields between two valleys are obtained,leading to the different out-of-plane spin relaxation times.The difference in the spin relaxation times can be suppressed by enhancing the intervalley hole-phonon scattering through increasing temperature or hole density.Particularly,in the relaxation of large spin polarization,it is interesting to discover that the initially equal hole densities in the two valleys are broken during the temporal evolution,and a valley polarization is built up.According to our calculation,with the spin polarization reaching 60%,the accessible valley polarization can reach beyond 1%and last hundreds of picoseconds,providing a large possibility for the experimental detection.The intravalley system in bilayer transition-metal dichalcogenides is a typical Rashba spin-orbit coupled one with the Zeeman field.The microscopic investigation on the spin diffusion in such a system is important,but the related study is still absent in the literature.For out-of-plane spins,we analytically show that the spin-diffusion process in each valley can be divided into four regimes by tuning the total effective magnetic field strength,and in each regime,the spin-diffusion length exhibits different dependencies on the scattering,total effective magnetic field and spin-orbit-coupling strengths.Then,the different total effective magnetic fields in the two valleys lead to different spin-diffusion lengths,whereas this difference in the spin-diffusion lengths can be suppressed by enhancing the intervalley hole-phonon scattering.Moreover,with a fixed large spin injection,we predict,the buildup of a steady-state valley polarization during the spin diffusion,similar to the induced valley polarization in the time domain.Nevertheless,with the increase of the impurity density to enhance intravalley scattering,the valley polarization in the time domain decreases whereas the one in the spatial domain gets enhanced.In the main part,from Chapter 3 to 11,focusing on the field of superconductors,we first study the equilibrium superconductivity in systems with the translational-symmetry breaking,and then,perform the investigation on the nonequilibrium electromagnetic re-sponse of superconductors.For the equilibrium study,in Chapter 3,we first introduce the symmetric classifications of the Cooper pair:even-frequency singlet,odd-frequency singlet,even-frequency triplet and odd-frequency triplet types.The required symmetry breakings to realize the last three unconventional Cooper pairs in homogeneous systems are also introduced.While the existence of the unconventional Cooper pair does not guarantee the generation of the unconventional superconductivity,since the latter also requires the special symmetry of pair potential.After that,we introduce the basic equa-tion of two-point Green function in superconductors:Gorkov equation,which contains all the information of system and can therefore be used as a starting point for studying and calculating the properties of superconductors.Within the equilibrium Gorkov equa-tion,we introduce several materials/systems for the possible realization of unconventional Cooper pair/superconductivity,including the conventional superconductors in proximity to ferromagnet,non-centrosymmetric superconductors,spin-orbit coupled convent ional su-perconductors,controversial unconventional superconductor Sr2RuO4,and supereonduct-ing heavy-fermion materials with possible p-wave attractive potential.The possibility to spntaneously break the translational symmetry in homogeneous Superconductors by us-ing Zeeman field,i.e.,Fulde-Ferrell-Larkin-Ovchinnikov(FFLO)state,is also introduced.Whereas the spontaneous breaking of the continuous rotational symeetry in isotropic sys-tem is dotrimental for FFLO state to survive the impurity defect and thermodynamic flue-tuation.Nevertheless,it is reported that by using the spin-orbit coupling,an anisotropie inismatch between spin-up and-down Fermi surfaces emerges with tho Zeeman ficld.and this anisotropy stabilizes the FFLO state through optimization of the center-of-mass mo-mentum of Cooper pair.The corresponding theoretical progress is then reviewed.In Chapter 4,by using the equilibrium Gorkov equation,we investigate the realization of the unconventional Cooper pairs and order parameters with the translational-symmetry breaking.Based on the symmetry analysis,we first discuss the requirement to realize unconventional Cooper pairs with the translational-symmetry breaking.It is found that with the translational-symmetry breaking,the odd-frequency singlet Cooper pair exists intrinsically,and both existences of the even-and odd-frequency triplet types only require the spin-rotational-symmetry breaking.Therefore,we point out that all four types of Cooper pair can be realized in spin-orbit-coupled quantum well in proximity to s-wave superconductors with the translational-symmetry breaking.With these pairings,it is shown that all four types of the order parameters can be obtained from the self-energy of the electron-electron Coulomb interaction with the inevitable plasmon effect in two-dimensional quantum well.By considering a specific case in InSb(110)quantum well in proximity to s-wave superconductors in FFLO phase or with a supercurrent,we derive the analytic expressions for all four types of the order parameters,and s-wave singlet even-frequency,p-wave singlet odd-frequency,p-wave triplet even-frequency and d-wave triplet odd-frequency order parameters are obtained due to the material property.We show that at proper density,the conventional s-wave order parameter is suppressed,and then,the unconventional ones dominate,providing the possibility for the experimental detection.In Chapter 5,we investigate the Fulde-Ferrell state in the spin-orbit-coupled s-wave superconductors.Differing from the full numerical approach for searching multi-variable minimum in the literature,we analytically derive the anomalous Green function and hence the gap equation by using the equilibrium Gorkov equation,and determine the supercon-ducting state by calculating the energy minimum with respect to a single parameter,i.e.,the center-of-mass momentum of Cooper pair.Then,the microscopic properties of the superconducting state can be well discussed.We show that in spin-orbit-coupled super-conductors,the presence of magnetic field leads to two superconducting phases with the center-of-mass momentum of Cooper pair.At small magnetic field,the center-of-mass momentum of Cooper pair is induced due to the energy-spectrum distortion and no un-pairing region with vanishing singlet correlation appears.We refer to this state as the drift-BCS state.By further increasing the magnetic field to a critical field,the observed abrupt enhancement of the center-of-mass momentum of Cooper pair and suppression on the order parameter indicate the occurrence of a first-order phase transition.After the transition,the superconducting state falls into the Fulde-Ferrell state with the emergence of the unpairing regions.Enhanced Pauli limit and hence enlarged magnetic-field regime of the Fulde-Ferrell state,which arise from the spin-flip terms of the spin-orbit coupling,are also revealed.By studying the triplet correlations induced by the spin-orbit coupling,we show that the Cooper-pair spin polarization exhibits different magnetic-field depen-dences between the drift-BCS and Fulde-Ferrell states,providing an experimental scheme to distinguish these two phases.From Chapter 6 to 11,we study therich electromagnetic properties of superconductiv-ity from the view of nonequilibrium dynamics.In Chapter 6,we first introduce the gauge structure in superconductors first revealed by Nambu and demonstrate that the gauge in-variance in superconducting states is equivalent to the charge conservation.Then,various collective excitations,including Nambu-Goldstone mode(phase fluctuation of the order parameter)and the related Anderson-Higgs mechanism,Leggett mode(phase-difference fluctuation of the order parameters in two-band superconductors),Carlson-Goldman mode(Nambu-Goldstone mode near Tc),Higgs mode(amplitude fluctuation of the order parame-ter)as well as Bardasis-Schrieffer mode(amplitude fluctuation of the order parameter with different angular momentum from the equilibrium order parameter),are introduced.We also introduce the impurity influence on the equilibrium property of superconductors,i.e.,Anderson theorem.After that,we review the experimental and theoretical progresses of the electromagnetic response and in particular,the THz optical response of superconductors.The reviewed experimental progress includes the Meissner effect in the stationary mag-netic response,macroscopic Ginzburg-Landau phenomenological theory for expermental analysis in the early-stage works,two-fluid model for describing the optical conductivity in low-frequency optical response,and the phenomena in the THz optical response(such as different optical absorption behaviors of superconductors in the normal-and anomalous-skin-effect regions,Higgs-mode excitation in the non-linear regime and a ?-jump of the phase in the corresponding signal,as well as Leggett-mode excitation in the non-linear regime in two-band superconductors).For the theoretical progress,we point out that in principle,a complete theory to calculate the electromagnetic properties must satisfy the following four condit ions:(?)it should be capable of calculating both magnetic and opti-cal responses in linear and nonlinear regimes,i.e.,it should include both electromagnetic effects by electric field E and directly by vector potential A;(?)it should be capable of cal-culating the electromagnetic response of the various collective excitations;(?)it should be capable of dealing with the scattering effect,which is inevitable in dirty superconducting metals;(?)it should satisfy the gauge invariance in superconductors(i.e.,gauge structure revealed by Nambu)that is very important in the superconducting state.Nevertheless,in contrast to the growing rich experimental findings,although the microscopic theories of the electromagnetic properties of superconductors within the framework of BCS superconduc-tivity theory have been developed for more than five decades,the theoretical descriptions,including the Mattis-Bardeen theory based on Kubo current-current corrlation approach in the anomalous-skin-effect region,Bloch or Liouville equation derived in the Anderson pseudo-spin picture,the semiclassical Boltzmann equation of quasiparticles,Eilenberger and Usadel equation derived from the basic Gorkov equation within the quasiclassical for-malism by using ?r3-Green function,and gauge-invariant optical Bloch equation established by Yu and Wu within the equal-time scheme by using ?0-Green function,are incapable of satisfying all conditions above,and hence,have certain deficiencies.In Chapter 7,by using the gauge-invariant optical Bloch equation approach,we per-form a kinetic investigation on the anomalous Hall effect in chiral p-wave superconducting states.We first prove that the intrinsic anomalous Hall conductivity is zero as a conse-quence of Galilean invariance,whereas the impurity scattering leads to the finite extrinsic one.In contrast to the well-established Kubo formalism for describing the conventional skew-scattering channel in the linear response,our study not only provides a microscopic kinetic description of this channel,but also reveals a new induction channel from the Born contribution by nonlinear optical excitation.This new channel has long been overlooked in the literature due to the difficulty in treating the quasiparticle correlation in the previous semiclassical approach or including the nonlinear effect in Kubo diagrammatic formalism,but it dominates the anomalous Hall current at the weak impurity interaction.Finally,inspired by recently observed penetrations of the superconductivity and exchange field in meal/ferromagnet/superconductor junction,we study the case with a transverse con-ical magnetization,which breaks the Galilean invariance.In this situation,the intrinsic anomalous Hall conductivity is no longer zero.In Chapter 8,we develop the gauge-invariant optical Bloch equation to include the complete superfluid dynamics and electromagnetic effects,and then,establish gauge-invariant kinetic equation of superconductors.We first prove that the gauge-invariant kinetic equation satisfies the gauge structure revealed by Nambu,and then,the retained gauge invariance directly leads to the charge conservation in the electromagnetic response.Through the gauge-invariant kinetic equation,we first focus on the current excitation in the magnetic and low-frequency optical responses.We predict that the normal fluid and hence the scatt ering are present only when the excited superconducting velocity ?s is larger than a threshold ?L=|?|/kF.Interestingly,we find that there exists friction between the normal-fluid and superfluid currents,and part of the superfluid becomes viscous due to this friction.Therefore,a three-fluid model:normal fluid,non-viscous and viscous super-fluids,is proposed for capturing the electromagnetic response of the superconducting state.Specifically,for the stationary magnetic response,at ?s<?L with only the non-viscous superfluid,we recover the Meissner supercurrent in the literature and the gap equation can exactly reduce to the Ginzburg-Landau equation near critical temperature.At ?s>?L,the magnetic response is captured by the three-fluid model.In addition to the directly excited Meissner supercurrent in the superfluid by magnetic flux,normal-fluid current is also induced through the friction drag with the viscous superfluid current.Due to the normal-fluid and viscous superfluid currents,the penetration depth is influenced by the scattering effect.Moreover,we also predict an exotic phase in which both the resistivity and superconducting gap are finite.As for the optical response,we show that the ex-cited normal-fluid current exhibits the Drude-model behavior while the superfluid current consists of the Meissner supercurrent and Bogoliubov quasiparticle current.Then,at low temperature,we exactly recover the two-fluid model in the literature.Nevertheless,due to the friction between the superfluid and normal-fluid currents,we show that the optical conductivity is captured by the three-fluid model.In Chapter 9,we show that the gauge-invariant kinetic equation of superconductors provides an efficient approach to calculate the electromagnetic responses of the Nambu-Goldstone and Higgs modes on an equal footing.Both linear and second-order optical responses are analytically investigated.The linear response agrees with previous results in the literature.Specifically,the linear response of the Higgs mode vanishes in the long-wave limit,and hence does not manifest itself in the optical experiments.Whereas the linear re-sponse of the Nambu-Goldstone mode interacts with the long-range Coulomb interaction,causing the original gapless energy spectrum effectively lifted up to the plasmon frequency as a result of the Anderson-Higgs mechanism,and hence,becomes inactive.The second-order response exhibits interesting physics.Oi one hand,a finite second-order optical response of the Higgs mode is obtained in the long-wave limit,and exhibits a resonance at 2?=2?,in consistency with the experimental findings.We reveal that this response is attributed solely to the drive effect(i.e.drive effect of optical electric field and diamag-netic effect of vector potential)rather than the widely considered Anderson-pump effect(paramagnetic effect of vector potential)in the literature.On the other hand,we also find a finite second-order response of the Nambu-Goldstone mode.Particularly,it is shown that this response decouples with the long-range Coulomb interaction as a consequence of the charge conservation and hence becomes free from the influence of the Anderson-Higgs mechanism.The Nambu-Goldstone mode therefore maintains the original gapless energy spectrum in the second-order optical response,and hence retains active.A tentative scheme based on the Josephson junction to detect this response is then proposed.In Chapter 10,by applying the gauge-invariant kinetic equation,we analytically in-vestigatc the scattering influence on the THz optical properties of superconductors in the normal-skin-effect region.Both linear and second-order responses are studied under a multi-cycele)terahertz pulse.The optical absorption ?1s(?)in the linear regime,induced by the scattering,can well capture theoxporimentally observed features indirty supercon-ductors lying in the normal-skin-effect region,including the erossover point at ?=2|?|and the upturn with decreasing the frequency for ?<2|?| atlow temperature.Moreover,when the order parameter tends to zero at T>TC,the optical conductivity from the gauge-invariant kinetic equation exactly recovers the one in normal metals as the Drude model or conventional Boltzmann equation revealed.To the best of our knowledge,so far there is no theory of the optical conductivity in the literature that can rigorously recover the conductivity in normal metals from T<Tc to T>Tc,due to the difficulty in calculat-ing the vertex correction in superconductors self-consistently.The gauge-invariant kinetic equation hence provides an efficient approach to deal with the scattering.In the second-order regime,we show that the scattering causes a phase shift in the optical response of the Higgs mode.Particularly,this phase shift exhibits a significant ?-jump at ?=|?|,which provides a very clear feature for the experimental detection.Finally,by studying the damping of the Higgs-mode excitation after the pulse,a relaxation mechanism from the elastic scattering is also revealed.In Chapter 11,by applying the gauge-invariant kinetic equation in d-wave supercon-ductors,we analytically derive the energy spectra of the breathing Higgs mode and,in particular,rotating Higgs mode that is unique for the d-wave order parameter.Analytical investigation on their dynamic properties is also presented.We show that the breathing Higgs mode is optically visible in the second-order regime,irrelevant of optical polarization direction.Whereas the rotating Higgs mode is optically inactive,we show that this mode responds to magnetic field in the linear regime,suggesting a possible detection by mag-netic resonance experiment in the pseudogap phase.Particularly,it is interesting to find that the charge-neutral rotating Higgs mode,which does not manifest itself in the elec-tric measurement,contributes to a negative thermal Hall conductivity in the pseudogap phase.This finding is likely to capture a very recent experimental observation of negative thermal Hall signal in cuprate superconductors for the overdoped pseudogap phase which has attracted much interest.We therefore suggest that the unidentified charge-neutral excitation in that experiment to generate thermal Hall signal in the pseudogap phase is the rotating Higgs mode.Finally,we summarize in Chapter 12.