The study of nonequilibrium phenomena in correlated lattice systems has developed into one of the most active and exciting branches of condensed matter physics. This research field provides rich new insights that could not be obtained from the study of equilibrium situations, and the theoretical understanding of the physics often requires the development of new concepts and methods. On the experimental side, ultrafast pump-probe spectroscopies enable studies of excitation and relaxation phenomena in correlated electron systems, while ultracold atoms in optical lattices provide a new way to control and measure the time evolution of interacting lattice systems with a vastly different characteristic time scale compared to electron systems. A theoretical description of these phenomena is challenging because, first, the quantum-mechanical time evolution of many-body systems out of equilibrium must be computed and second, strong-correlation effects which can be of a nonperturbative nature must be addressed. This review discusses the nonequilibrium extension of the dynamical mean field theory (DMFT), which treats quantum fluctuations in the time domain and works directly in the thermodynamic limit. The method reduces the complexity of the calculation via a mapping to a selfconsistent impurity problem, which becomes exact in infinite dimensions. Particular emphasis is placed on a detailed derivation of the formalism, and on a discussion of numerical techniques, which enable solutions of the effective nonequilibrium DMFT impurity problem. Insights gained into the properties of the infinite-dimensional Hubbard model under strong nonequilibrium conditions are summarized. These examples illustrate the current ability of the theoretical framework to reproduce and understand fundamental nonequilibrium phenomena, such as the dielectric breakdown of Mott insulators, photodoping, and collapse-and-revival oscillations in quenched systems. Furthermore, remarkable novel phenomena have been predicted by the nonequilibrium DMFT simulations of correlated lattice systems, including dynamical phase transitions and field-induced repulsion-to-attraction conversions.

Superconductors host collective modes that can be manipulated with light. We show that a strong terahertz light field can induce oscillations of the superconducting order parameter in NbN with twice the frequency of the terahertz field. The result can be captured as a collective precession of Anderson's pseudospins in ac driving fields. A resonance between the field and the Higgs amplitude mode of the superconductor then results in large terahertz third-harmonic generation. The method we present here paves a way toward nonlinear quantum optics in superconductors with driving the pseudospins collectively and can be potentially extended to exotic superconductors for shedding light on the character of order parameters and their coupling to other degrees of freedom.

We construct a systematic high-frequency expansion for periodically driven quantum systems based on the Brillouin-Wigner (BW) perturbation theory, which generates an effective Hamiltonian on the projected zero-photon subspace in the Floquet theory, reproducing the quasienergies and eigenstates of the original Floquet Hamiltonian up to desired order in 1/omega, with omega being the frequency of the drive. The advantage of the BW method is that it is not only efficient in deriving higher-order terms, but even enables us to write down the whole infinite series expansion, as compared to the van Vleck degenerate perturbation theory. The expansion is also free from a spurious dependence on the driving phase, which has been an obstacle in the Floquet-Magnus expansion. We apply the BW expansion to various models of noninteracting electrons driven by circularly polarized light. As the amplitude of the light is increased, the system undergoes a series of Floquet topological-to-topological phase transitions, whose phase boundary in the high-frequency regime is well explained by the BW expansion. As the frequency is lowered, the high-frequency expansion breaks down at some point due to band touching with nonzero-photon sectors, where we find numerically even more intricate and richer Floquet topological phases spring out. We have then analyzed, with the Floquet dynamical mean-field theory, the effects of electron-electron interaction and energy dissipation. We have specifically revealed that phase transitions from Floquet-topological to Mott insulators emerge, where the phase boundaries can again be captured with the high-frequency expansion

A superconductor illuminated by an ac electric field with frequency Ω is theoretically found to generate a collective precession of Anderson's pseudospins, and hence a coherent amplitude oscillation of the order parameter, with a doubled frequency 2Ω through a nonlinear light-matter coupling. We provide a fundamental theory, based on the mean-field formalism, to show that the induced pseudospin precession resonates with the Higgs amplitude mode of the superconductor at 2Ω = 2∆ with 2∆ being the superconducting gap. The resonant precession is accompanied by a divergent enhancement of the third-harmonic generation (THG). By decomposing the THG susceptibility into the bare one and vertex correction, we find that the enhancement of the THG cannot be explained by individual quasiparticle excitations (pair breaking), so that the THG serves as a smoking gun for an identification of the collective Higgs mode. We further explore the effect of electronelectron scattering on the pseudospin resonance by applying the nonequilibrium dynamical mean-field theory to the attractive Hubbard model driven by ac electric fields. The result indicates that the pseudospin resonance is robust against electron correlations, although the resonance width is broadened due to electron scattering, which determines the lifetime of the Higgs mode.

We propose to combine the Floquet formalism for systems in ac fields with the dynamical mean-field theory to study correlated electron systems periodically driven out of equilibrium by external fields such as intense laser light. This approach has a virtue that we can nonperturbatively include both the correlation effects and nonlinear effects due to the driving field, which is imperative in analyzing recent experiments for photoinduced phase transitions. In solving the problem, we exploit a general theorem that the Hamiltonian in a Floquet matrix form can be exactly diagonalized for single-band noninteracting systems. As a demonstration, we have applied the method to the Falicov-Kimball model in intense ac fields to calculate the spectral function. The result shows that photoinduced midgap states emerge from strong ac fields, triggering an insulator-metal transition.

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We study dynamical phase transitions from antiferromagnetic to paramagnetic states driven by an interaction quench in the fermionic Hubbard model using the nonequilibrium dynamical mean-field theory. We identify two dynamical transition points where the relaxation behavior qualitatively changes: one corresponds to the thermal phase transition at which the order parameter decays critically slowly in a power law ∝ t −1/2 , and the other is connected to the existence of nonthermal antiferromagnetic order in systems with effective temperature above the thermal critical temperature. The frequency of the amplitude mode extrapolates to zero as one approaches the nonthermal (quasi)critical point, and thermalization is significantly delayed by the trapping in the nonthermal state. A slow relaxation of the nonthermal order is followed by a faster thermalization process. PACS numbers: 71.10.Fd, 64.60.Ht In many physical systems out of equilibrium, phase transitions occur as a real-time process of symmetry breaking or symmetry recovery. Examples for such "dynamical phase transitions" include the evolution of the Universe [1], liquid helium [2], and photoinduced phase transition in solids [3][4][5]. The macroscopic aspects are often described by the timedependent Ginzburg-Landau theory, where the order parameter is supposed to vary sufficiently slowly in time and space, so that the system can be considered to be locally close to thermal equilibrium. On the other hand, recent experimental developments of time-resolved measurement techniques in solids [6] and cold atoms [7] allow one to study dynamical phase transitions very far from equilibrium on the microscopic time scale of correlated quantum systems. In these cases, a "near-equilibrium" description might not be applicable. For instance, it has been recently suggested that superconductivity can be induced above the equilibrium critical temperature (T c ) by coherently exciting certain lattice vibrations, and that it lasts for a relatively long time (a few tens of ps) before thermalization occurs [5]. This observation is reminiscent of the prethermalization phenomenon [8][9][10][11], or the dynamics in the presence of a nonthermal fixed point in relativistic quantum field theories [12]. A fundamental question that we pose here is if the existence of such a nonthermal fixed point in correlated condensed matter systems allows symmetry broken states to survive above T c , and how it affects the dynamics.An important and still unresolved issue is how to characterize a nonequilibrium phase transition and its critical behavior for quantum systems [13,14]. Previous studies have in particular focused on the dynamics near quantum phase transitions in low dimensional systems (e. g., Refs. [15][16][17][18]). Higher dimensional systems are usually expected to show a thermal criticality out of equilibrium since quantum fluctuations are well suppressed. In this Letter, we study a dynamical phase transition for a simple microscopic model of correlated materials, namely the Hubbard mode...

We investigate the terahertz (THz)-pulse-driven nonlinear response in the d-wave cuprate superconductor Bi_{2}Sr_{2}CaCu_{2}O_{8+x} (Bi2212) using a THz pump near-infrared probe scheme in the time domain. We observe an oscillatory behavior of the optical reflectivity that follows the THz electric field squared and is markedly enhanced below T_{c}. The corresponding third-order nonlinear effect exhibits both A_{1g} and B_{1g} symmetry components, which are decomposed from polarization-resolved measurements. A comparison with a BCS calculation of the nonlinear susceptibility indicates that the A_{1g} component is associated with the Higgs mode of the d-wave order parameter.