Discussion Of Numerical Techniques Research Articles

Nonequilibrium dynamical mean-field theory and its applications

The study of nonequilibrium phenomena in correlated lattice systems has developed into an active and exciting branch 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, ultra-fast pump-probe spectroscopies enable studies of excitation and relaxation phenomena in correlated electron systems, while ultra-cold atoms in optical lattices provide a new way to control and measure the time-evolution of interacting lattice systems with a vastly different characteristic timescale compared to electron systems. A theoretical description of these phenomena is challenging because, firstly, we have to compute the quantum-mechanical time-evolution of many-body systems out of equilibrium, and secondly, deal with strong-correlation effects which can be of nonperturbative nature. In this review, we discuss 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 self-consistent impurity problem. 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. We summarize the insights gained into the properties of the infinite-dimensional Hubbard model under strong non-equilibrium conditions. 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, photo-doping, and collapse-and-revival oscillations in quenched systems.

Quantum Physics in One Dimension

Quantum Physics in One Dimension

Time-domain elastodynamic scattering problems

The interaction of an ultrasonic pulse with an inhomogeneity in an elastic solid is generally analyzed by the use of the FFT over time in combination with a numerical method for the scattering problem in the frequency domain. For pulses with a high-frequency content this approach tends to be computationally intensive. In this talk the possibilities of a direct analytical time-domain approach are explored, and then there is a discussion of numerical techniques. Results are presented for a single crack (slit and penny-shaped) and for macrocrack-microcrack configurations. Some of these results have been obtained by the finite difference method. The main emphasis in this paper is, however, on the development of a time-domain boundary integral equation method. By the use of appropriate representation integrals, a system of boundary integral equations has been obtained, which has subsequently been cast in a form that is amenable to a solution by the boundary element method in conjunction with a time-stepping technique. Particular attention has been devoted to dynamic overshoots of the stress intensity factors. Elastodynamic stress intensity factors have been computed as functions of time, and they have been compared with results of other authors.