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.

We use nonequilibrium dynamical mean-field theory to study the time evolution of the fermionic Hubbard model after an interaction quench. Both in the weak-coupling and in the strong-coupling regime the system is trapped in quasistationary states on intermediate time scales. These two regimes are separated by a sharp crossover at U(c)dyn=0.8 in units of the bandwidth, where fast thermalization occurs. Our results indicate a dynamical phase transition which should be observable in experiments on trapped fermionic atoms.

A quantum many-body system which is prepared in the ground state of an integrable Hamiltonian does not directly thermalize after a sudden small parameter quench away from integrability. Rather, it will be trapped in a prethermalized state and can thermalize only at a later stage. We discuss several examples for which this prethermalized state shares some properties with the nonthermal steady state that emerges in the corresponding integrable system. These examples support the notion that nonthermal steady states in integrable systems may be viewed as prethermalized states that never decay further. Furthermore we show that prethermalization plateaus are under certain conditions correctly predicted by generalized Gibbs ensembles, which are the appropriate extension of standard statistical mechanics in the presence of many constants of motion. This establishes that the relaxation behaviors of integrable and nearly integrable systems are continuously connected and described by the same statistical theory.

The strongest interaction between microscopic spins in magnetic materials is the exchange interaction Jex. Therefore, ultrafast control of Jex holds the promise to control spins on ultimately fast timescales. We demonstrate that time-periodic modulation of the electronic structure by electric fields can be used to reversibly control Jex on ultrafast timescales in extended antiferromagnetic Mott insulators. In the regime of weak driving strength, we find that Jex can be enhanced and reduced for frequencies below and above the Mott gap, respectively. Moreover, for strong driving strength, even the sign of Jex can be reversed and we show that this causes time reversal of the associated quantum spin dynamics. These results suggest wide applications, not only to control magnetism in condensed matter systems, for example, via the excitation of spin resonances, but also to assess fundamental questions concerning the reversibility of the quantum many-body dynamics in cold atom systems.

We present the exact solution of the Falicov-Kimball model after a sudden change of its interaction parameter using non-equilibrium dynamical mean-field theory. For different interaction quenches between the homogeneous metallic and insulating phases the system relaxes to a non-thermal steady state on time scales on the order of /bandwidth, showing collapse and revival with an approximate period of h/interaction if the interaction is large. We discuss the reasons for this behavior and provide a statistical description of the final steady state by means of generalized Gibbs ensembles.PACS numbers: 03.75. Ss, 05.30.Fk, 71.27.+a How does an isolated quantum-mechanical many-body system develop after it is suddenly forced out of thermal equilibrium? Under which conditions does it relax to a new steady state, and how fast? Is it ergodic so that it reaches a new thermodynamic equilibrium, or does the final state depend on the initial state? Recently it has become feasible to study these fundamental questions experimentally and theoretically. In experiments with ultracold atomic gases [1] it is possible to subject a prepared initial state to a rapid change of system parameters. Long observation times are possible due to the excellent isolation from the environment. For example, Bose-Einstein condensates (BECs) were quenched across the superfluid-insulator transition and back [2], their collapse and revival after a quench was observed [3], a quenched spinor BEC was found to exhibit spontaneous symmetry breaking [4], and a quantum version of Newton's cradle was found not to thermalize [5].One might expect that a quenched system with many interacting degrees of freedom will relax to a new thermal state, characterized only by a few thermodynamic variables such as internal energy and particle number. However this may not be the case if the system is integrable, because then the final state is constrained by infinitely many constants of motion. Indeed, theoretical studies for one-dimensional hard-core bosons [6,7] (experimentally realized in Ref. 5) and for the Luttinger model [8] found that these integrable systems relax to non-thermal steady states. Nevertheless for both models the final state is described by a generalized Gibbs ensemble [6], which maximizes the entropy subject to all constraints. On the other hand, for non-integrable and unconstrained systems the usual Gibbs ensemble should describe the final steady state. In contrast to this expectation recent numerical studies for finite one-dimensional systems of soft-core bosons [9] and spinless fermions [10] did not find thermalization. While the reasons for this behavior are not yet understood, hard-core bosons in two dimensions do thermalize as expected [11]. Clearly finitesize effects must be well-controlled in all such calculations in order to obtain the correct behavior at large times.Dynamical mean-field theory (DMFT) [12,13], which has become a standard technique for correlated systems in equilibrium, can also provide insight into their quantum dynamics, e....

We solve the impurity problem which arises within nonequilibrium dynamical mean-field theory for the Hubbard model by means of a self-consistent perturbation expansion around the atomic limit. While the lowest order, known as the non-crossing approximation (NCA), is reliable only when the interaction U is much larger than the bandwidth, low-order corrections to the NCA turn out to be sufficient to reproduce numerically exact Monte Carlo results in a wide parameter range that covers the insulating phase and the metal-insulator crossover regime at not too low temperatures. As an application of the perturbative strong-coupling impurity solver we investigate the response of the double occupancy in the Mott insulating phase of the Hubbard model to a dynamical change of the interaction or the hopping, a technique which has been used as a probe of the Mott insulating state in ultracold fermionic gases.

Using nonequilibrium dynamical mean-field theory, we compute the time evolution of the current in a Mott insulator after a strong electric field is turned on. We observe the formation of a quasistationary state in which the current is almost time independent although the system is constantly excited. At moderately strong fields this state is stable for quite long times. The stationary current exhibits a threshold behavior as a function of the field, in which the threshold increases with the Coulomb interaction and vanishes as the metal-insulator transition is approached.

We present optimized implementations of the weak-coupling continuous-time Monte Carlo method defined for nonequilibrium problems on the Keldysh contour. We describe and compare two methods of preparing the system before beginning the real-time calculation: the "interaction quench" and the "voltage quench", which are found to be suitable for large and small voltage biasses, respectively. We also discuss technical optimizations which increase the efficiency of the real-time measurements. The methods allow the accurate simulation of transport through quantum dots over wider interaction ranges and longer times than have heretofore been possible. The current-voltage characteristics of the particle-hole symmetric Anderson impurity model is presented for interactions U up to 10 times the intrinsic level width Γ. We compare the Monte Carlo results to fourth order perturbation theory, finding that perturbation theory begins to fail at U/Γ 4. Within the parameter range studied we find no evidence for a splitting of the Kondo resonance due to the applied voltage. The interplay of voltage and temperature and the Coulomb blockade conductance regime are studied.