arXiv:1702.01956 · Physical review. B./Physical review. B
Nonequilibrium dynamics in the one-dimensional Fermi-Hubbard model: Comparison of the nonequilibrium Green-functions approach and the density matrix renormalization group method
Niclas Schlünzen, Jan‐Philip Joost, Fabian Heidrich‐Meisner, M. Bönitz
Experiments with ultracold fermionic atoms in optical lattices have attracted much interest in condensed matter physics as they serve as a model for electrons in solids. While experiments now allow the observation of nonequilibrium transport processes with single-site resolution, the theoretical description, especially for strongly correlated fermions, remains very challenging. Standard methods, such as the density matrix renormalization group (DMRG) and nonequilibrium Green functions (NEGF), have been primarily applied to one-dimensional systems and weak to moderate coupling, respectively. Here, the authors perform a detailed comparison of DMRG and NEGF and demonstrate that both can be benchmarked against each other. They observe complementary applicability ranges suggesting that a combination of both will allow to substantially expand the range and duration of first-principles quantum dynamics simulations for fermionic lattice systems.
Experiments with ultracold fermionic atoms in optical lattices have attracted much interest in condensed matter physics as they serve as a model for electrons in solids. While experiments now allow the observation of nonequilibrium transport processes with single-site resolution, the theoretical description, especially for strongly correlated fermions, remains very challenging. Standard methods, such as the density matrix renormalization group (DMRG) and nonequilibrium Green functions (NEGF), have been primarily applied to one-dimensional systems and weak to moderate coupling, respectively. Here, the authors perform a detailed comparison of DMRG and NEGF and demonstrate that both can be benchmarked against each other. They observe complementary applicability ranges suggesting that a combination of both will allow to substantially expand the range and duration of first-principles quantum dynamics simulations for fermionic lattice systems.
This article presents an overview on recent progress in the theory of nonequilibrium Green functions (NEGF). We discuss applications of NEGF simulations to describe the femtosecond dynamics of various finite fermionic systems following an excitation out of equilibrium. This includes the expansion dynamics of ultracold atoms in optical lattices following a confinement quench and the excitation of strongly correlated electrons in a solid by the impact of a charged particle. NEGF, presently, are the only ab initio quantum approach that is able to study the dynamics of correlations for long times in two and three dimensions. However, until recently, NEGF simulations have mostly been performed with rather simple selfenergy approximations such as the second-order Born approximation (SOA). While they correctly capture the qualitative trends of the relaxation towards equilibrium, the reliability and accuracy of these NEGF simulations has remained open, for a long time. Here we report on recent tests of NEGF simulations for finite lattice systems against exact-diagonalization and density-matrix-renormalization-group benchmark data. The results confirm the high accuracy and predictive capability of NEGF simulations-provided selfenergies are used that go beyond the SOA and adequately include strong correlation and dynamical-screening effects. With an extended arsenal of selfenergies that can be used effectively, the NEGF approach has the potential of becoming a powerful simulation tool with broad areas of new applications including strongly correlated solids and ultracold atoms. The present review aims at making such applications possible. To this end we present a selfcontained introduction to the theory of NEGF and give an overview on recent numerical applications to compute the ultrafast relaxation dynamics of correlated fermions. In the second part we give a detailed introduction to selfenergies beyond the SOA. Important examples are the third-order approximation, the [Formula: see text] approximation, the T-matrix approximation and the fluctuating-exchange approximation. We give a comprehensive summary of the explicit selfenergy expressions for a variety of systems of practical relevance, starting from the most general expressions (general basis) and the Feynman diagrams, and including also the important cases of diagonal basis sets, the Hubbard model and the differences occuring for bosons and fermions. With these details, and information on the computational effort and scaling with the basis size and propagation duration, readers will be able to choose the proper basis set and straightforwardly implement and apply advanced selfenergy approximations to a broad class of systems.
Correlated real-time dynamics in large, spatially inhomogeneous quantum systems remain difficult to access with nonequilibrium many-body methods. Two-time nonequilibrium Green functions (NEGF) retain dynamical correlations but their computational runtime grows cubically with the number of time steps Nt. This scaling bottleneck could recently be overcome by introducing the G1--G2 scheme that is linear in Nt, but requires propagation of a two-particle correlation function and may suffer from numerical instabilities. This has restricted simulations to small systems with Nb∼102 basis states. Here we introduce a quantum-fluctuation formulation of nonequilibrium Green functions, denoted δNEGF, that represents dynamical two-particle correlations through fluctuations of field-operator products, δG^. This guarantees stable dynamics by preserving the positivity of the reduced density matrices, avoids the explicit storage of the two-particle Green function, and reduces the propagation to a finite ensemble of Hartree-Fock-like trajectories. Combined with a stochastic low-rank decomposition of the correlation functions, the method retains time-linear scaling while extending dynamical GW and particle-particle and particle-hole T-matrix simulations to basis sizes of order Nb∼104. We benchmark δNEGF against exact and HF-GKBA results for lattice systems, finding stable correlated dynamics also at strong coupling. We further demonstrate large-scale simulations of diffusion in two-dimensional Hubbard lattices and ultrafast relaxation in graphene nanoribbon heterostructures with long-range Coulomb interactions. These results establish δNEGF as a scalable route to dynamical self-energy simulations of large, spatially inhomogeneous correlated quantum systems beyond the reach of existing NEGF implementations.
The framework of nonequilibrium Green functions (NEGFs) is developed to accurately describe interacting quantum systems evolving in the presence of strongly exciting ultra-fast fields.At its core stands the selfconsistent determination of the selfenergy, a functional of the single-particle Green function, which incorporates all relevant many-body physics and, via diagrammatic techniques, guarantees the conservation of the constants of motion.Over the last decades, NEGFs have been successfully applied to a broad range of domains ranging from homogeneous systems such as non-ideal quantum gases and plasmas over condensed matter to highly inhomogeneous systems such as atoms and molecules photo-ionized by laser-fields or colliding nuclear matter.Only recently, the first accurate simulation of the expansion dynamics of clouds of ultra-cold fermionic atoms in a two-dimensional optical lattice could be achieved using NEGFs.The single-particle Green function is the solution of a coupled set of kinetic equations.Its intrinsic dependence on two independent time-arguments, ranging over the complete past of the system, not only enables the determination of all singleparticle observables but also provides access to important two-particle observables and spectral information.With this coupling to the history, though, comes the practical necessity to apply physically motivated approximation techniques to the structure of the selfenergy and the Green functions, to achieve numerical tractability.This thesis is devoted to provide a thorough overview of the possible approximation techniques including perturbative and non-perturbative approaches.Starting from the exact coupled set of equations of motion for the Green functions, the expressions for all selfenergy approximations are derived in the most general basis representation, i.e., for interaction tensor elements w ijkl .From this, the corresponding expressions in a diagonal basis, where the interaction can be described as a matrix w ij , and for the special case of the Hubbard model, where the interaction is just given as a scalar U , are deduced.
ABSTRACT We benchmark nonequilibrium Green's function (NEGF) approaches for interaction quenches in the half‐filled Fermi–Hubbard model in one and two dimensions. We compare fully self‐consistent two‐time Kadanoff–Baym equations (KBE), the generalized Kadanoff–Baym ansatz (GKBA), and the recently developed NEGF‐based quantum fluctuations approach (NEGF‐QF) with second‐order Born (2B) and GW self‐energies. As references, we use DMRG in 1D and time‐dependent perturbation theory at weak coupling. While all methods reproduce the early‐time build‐up of correlations (prethermalization), different deficiencies emerge during the slower relaxation in the long‐time limit: KBE shows overdamped oscillations, GKBA exhibits a negative overshoot of the Fermi‐surface jump, and NEGF‐QF develops a prethermalization plateau with delayed relaxation. In 1D, we find that GKBA + 2B agrees best with DMRG across the explored parameter range, while QF + GW deviates the most, consistent with its reduced self‐consistency. In 2D, all approaches correctly capture the momentum‐dependence of the relaxation dynamics.
The dynamics of strongly correlated fermions following an external excitation reveals extremely rich collective quantum effects. Examples are fermionic atoms in optical lattices, electrons in correlated materials, and dense quantum plasmas. Presently, the only quantum-dynamics approach that rigorously describes these processes in two and three dimensions is the nonequilibrium Green functions (NEGF) method. However, NEGF simulations are computationally expensive due to their T^{3} scaling with the simulation duration T. Recently, T^{2} scaling was achieved with the generalized Kadanoff-Baym ansatz (GKBA), for second-order Born (SOA) selfenergies, which has substantially extended the scope of NEGF simulations. Here we demonstrate that GKBA-NEGF simulations can be performed with order T^{1} scaling, both for SOA and GW selfenergies, and point out the remarkable capabilities of this approach.