Nonequilibrium Green Functions Simulations on the Next Level: Theoretical Advances and Applications to Finite Lattice Systems

This thesis is devoted to the description of correlated finite lattice systems under nonequilibrium conditions. In this context, the lack of small parameters in the corresponding standard many-body equations makes it difficult to construct suitable approximations for theoretical tools, which renders the computation of relevant observables numerically costly and impractical. At the same time, rigorous predictions for the ultrafast dynamics in correlated lattices are highly valuable for the understanding of many state-of-the-art experiments. The nonequilibrium Green functions (NEGF) technique is particularly well-suited to meet the challenging demands that come with the description of the nontrivial interplay between quantum correlations and nonequilibrium effects in excited lattice systems. However, in order to apply the approach on a practically relevant scale, several methodological improvements come to be indispensable. The present thesis contains these theoretical advances of the NEGF method, alongside with—thus accessible—applications to ultracold atoms in optical lattices and excited finite graphene nanostructures

Löwdin's symmetry dilemma within Green functions theory for the one‐dimensional Hubbard model

The energy gap of correlated Hubbard clusters is well studied for one-dimensional systems using analytical methods and density-matrix- renormalization-group (DMRG) simulations. Beyond 1D, however, exact results are available only for small systems by quantum Monte Carlo. For this reason and, due to the problems of DMRG in simulating 2D and 3D systems, alternative methods such as Green functions combined with many-body approximations (GFMBA), that do not have this restriction, are highly important. However, it has remained open whether the approximate character of GFMBA simulations prevents the computation of the Hubbard gap. Here we present new GFMBA results that demonstrate that GFMBA simulations are capable of producing reliable data for the gap which agrees well with the DMRG benchmarks in 1D. An interesting observation is that the accuracy of the gap can be significantly increased when the simulations give up certain symmetry restriction of the exact system, such as spin symmetry and spatial homogeneity. This is seen as manifestation and generalization of the “symmetry dilemma” introduced by Löwdin for Hartree–Fock wave function calculations

Accelerating Nonequilibrium Green functions simulations with embedding selfenergies

Real-time nonequilibrium Green functions (NEGF) have been very successful to simulate the dynamics of correlated many-particle systems far from equilibrium. However, NEGF simulations are computationally expensive since the effort scales cubically with the simulation duration. Recently we have introduced the G1--G2 scheme that allows for a dramatic reduction to time-linear scaling [Schl\"unzen, Phys. Rev. Lett. 124, 076601 (2020); Joost et al., Phys. Rev. B 101, 245101 (2020)]. Here we tackle another problem: the rapid growth of the computational effort with the system size. In many situations where the system of interest is coupled to a bath, to electric contacts or similar macroscopic systems for which a microscopic resolution of the electronic properties is not necessary, efficient simplifications are possible. This is achieved by the introduction of an embedding selfenergy -- a concept that has been successful in standard NEGF simulations. Here, we demonstrate how the embedding concept can be introduced into the G1--G2 scheme, allowing us to drastically accelerate NEGF embedding simulations. The approach is compatible with all advanced selfenergies that can be represented by the G1--G2 scheme [as described in Joost et al., Phys. Rev. B 105, 165155 (2022)] and retains the memory-less structure of the equations and their time linear scaling. As a numerical illustration we investigate the charge transfer between a Hubbard nanocluster and an additional site which is of relevance for the neutralization of ions in matter

Doublon production in correlated materials by multiple ion impacts

In a recent Letter [Balzer \textit{et al.}, Phys. Rev. Lett. \textbf{121}, 267602 (2018)] it was demonstrated that ions impacting a correlated graphene cluster can excite strongly nonequilibrium states. In particular, this can lead to an enhanced population of bound pairs of electrons with opposite spin -- doublons -- where the doublon number can be increased via multiple ion impacts. These predictions were made based on nonequilibrium Green functions (NEGF) simulations allowing for a time-dependent non-perturbative study of the energy loss of charged particles penetrating a strongly correlated system. Here we extend these simulations to larger clusters and longer simulation times, utilizing the recently developed G1--G2 scheme [Sch\"unzen \textit{et al.}, Phys. Rev. Lett. \textbf{124}, 076601 (2020)] which allows for a dramatic speedup of NEGF simulations. Furthermore, we investigate the dependence of the energy and doublon number on the time interval between ion impacts and on the impact point