The nonequilibrium description of quantum systems requires, for more than two or three particles, the use of a reduced description to be numerically tractable. Two possible approaches are based on either reduced density matrices or nonequilibrium Green functions (NEGF). Both concepts are formulated in terms of hierarchies of coupled equations—the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy for the reduced density operators and the Martin-Schwinger-hierarchy (MS) for the Green functions, respectively. In both cases, similar approximations are introduced to decouple the hierarchy, yet still many questions regarding the correspondence of both approaches remain open. Here we analyze this correspondence by studying the generalized Kadanoff–Baym ansatz (GKBA) that reduces the NEGF to a single-time theory. Starting from the BBGKY-hierarchy we present the approximations that are necessary to recover the GKBA result both, with Hartree-Fock propagators (HF-GKBA) and propagators in second Born approximation. To test the quality of the HF-GKBA, we study the dynamics of a 4-electron Hubbard nanocluster starting from a strong nonequilibrium initial state and compare to exact results and the Wang-Cassing approximation to the BBGKY hierarchy presented recently by Akbari et al. [1].
Abstract Correlated classical and quantum many‐particle systems out of equilibrium are of high interest in many fields, including dense plasmas, correlated solids, and ultracold atoms. Accurate theoretical description of these systems is challenging both, conceptionally and with respect to computational resources. While for classical systems, in principle, exact simulations are possible via molecular dynamics, this is not the case for quantum systems. Alternatively, one can use many‐particle approaches such as hydrodynamics, kinetic theory, or nonequilibrium Green functions (NEGF). However, NEGF exhibit a very unfavorable cubic scaling of the CPU time with the number of time steps. An alternative is the G1–G2 scheme [N. Schlünzen et al., Phys. Rev. Lett. 124, 076601 (2020)] which allows for NEGF simulations with time linear scaling, however, at the cost of large memory consumption. The reason is the need to store the two‐particle correlation function. This problem can be overcome for a number of approximations by reformulating the kinetic equations in terms of fluctuations – an approach that was developed, for classical systems, by Yu.L. Klimontovich [JETP 33, 982 (1957)]. Here, we present an overview of his ideas and extend them to quantum systems. In particular, we demonstrate that this quantum fluctuations approach can reproduce the nonequilibrium GW approximation [E. Schroedter et al., Cond. Matt. Phys. 25, 23401 (2022)] promising high accuracy at low computational cost which arises from an effective semiclassical stochastic sampling procedure. We also demonstrate how to extend the approach to the two‐time exchange‐correlation functions and the density response properties. [E. Schroedter et al., Phys. Rev. B 108, 205109 (2023)]. The results are equivalent to the Bethe–Salpeter equation for the two‐time exchange‐correlation function when the generalized Kadanoff‐Baym ansatz with Hartree‐Fock propagators is applied [E. Schroedter and M. Bonitz, phys. stat. sol. (b) 2024, 2300564].
The primary goals of this dissertaion are to describe and elucidate a new formalism to study the out of equilibrium dynamics of integrable models, and to apply this formalism to some specific problems. In particular, I describe the “quench dynamics” of a gas of interacting bosons in one dimension, and provide details of some preliminary work on the isotropic Heisenberg chain. The formalism, based on earlier work by Yudson, provides a way of getting around some of the difficulties involved in calculating the time evolution of arbitrary initial states evolved with integrable Hamiltonians. In addition to this, I aim to discuss some of the general aspects of quench dynamics in quantum systems. The description of the nonequilibrium dynamics of a given system depends significantly on initial states, and the time scales at which the system is probed compared with the various inherent time scales in the system. I present the experimental context and motivation for these studies, and survey the existing techniques and efforts at understanding the relaxation of systems that are far from their equilibrium states or ground states.