S. Hermanns

Dynamics of strongly correlated fermions: Ab initio results for two and three dimensions

Quantum transport of strongly correlated fermions is of central interest in condensed matter physics. Here, we present first-principle nonequilibrium Green functions results using T -matrix selfenergies for finite Hubbard clusters of dimension 1, 2, 3. We compute the expansion dynamics following a potential quench and predict its dependence on the interaction strength and particle number. We discover a universal scaling, allowing an extrapolation to infinite-size systems, which shows excellent agreement with recent cold atom diffusion experiments [Schneider et al., Nat. Phys. 8, 213 (2012)].

Electronic double excitations in quantum wells: Solving the two-time Kadanoff-Baym equations

For a quantum many-body system, the direct population of states of double-excitation character is a clear indication that correlations importantly contribute to its nonequilibrium properties. We analyze such correlation-induced transitions by propagating the nonequilibrium Greens functions in real time within the second Born approximation. As crucial benchmarks, we compute the absorption spectrum of few electrons confined in quantum wells of different width. Our results include the full two-time solution of the Kadanoff-Baym equations as well as of their time-diagonal limit and are compared to Hartree-Fock and exact diagonalization data.

The generalized Kadanoff-Baym ansatz. Computing nonlinear response properties of finite systems

For a minimal Hubbard-type system at different interaction strengths U, we investigate the density-response for an excitation beyond the linear regime using the generalized Kadanoff-Baym ansatz (GKBA) and the second Born (2B) approximation. We find strong correlation features in the response spectra and establish the connection to an involved double excitation process. By comparing approximate and exact Greens function results, we also observe an anomalous U-dependence of the energy of this double excitation in 2B+GKBA. This is in accordance with earlier findings [K. Balzer et al., EPL 98, 67002 (2012)] on double excitations in quantum wells.

The non-equilibrium Green function approach to inhomogeneous quantum many-body systems using the generalized Kadanoff–Baym ansatz

In the non-equilibrium Green function calculations, the use of the generalized Kadanoff–Baym ansatz (GKBA) allows for a simple approximate reconstruction of the two-time Green function from its time-diagonal value. With this, a drastic reduction of the computational needs is achieved in time-dependent calculations, making longer time propagation possible and more complex systems accessible. This paper gives credit to the GKBA that was introduced 25 years ago. After a detailed derivation of the GKBA, we recall its application to homogeneous systems and show how to extend it to strongly correlated, inhomogeneous systems. As a proof of concept, we present the results for a two-electron quantum well, where the correct treatment of the correlated electron dynamics is crucial for a correct description of the equilibrium and dynamic properties.

Hubbard nanoclusters far from equilibrium

The Hubbard model is a prototype for strongly correlated many-particle systems, including electrons in condensed matter and molecules, as well as for fermions or bosons in optical lattices. While the equilibrium properties of these systems have been studied in detail, the nonequilibrium dynamics following a strong non-perturbative excitation only recently came into the focus of experiments and theory. It is of particular interest how the dynamics depend on the coupling strength and on the particle number and whether there exist universal features in the time evolution. Here, we present results for the dynamics of finite Hubbard clusters based on a selfconsistent nonequilibrium Green functions (NEGF) approach invoking the generalized Kadanoff--Baym ansatz (GKBA). We discuss the conserving properties of the GKBA with Hartree--Fock propagators in detail and present a generalized form of the energy conservation criterion of Baym and Kadanoff for NEGF. Furthermore, we demonstrate that the HF-GKBA cures some artifacts of prior two-time NEGF simulations. Besides, this approach substantially speeds up the numerical calculations and thus presents the capability to study comparatively large systems and to extend the analysis to long times allowing for an accurate computation of the excitation spectrum via time propagation. Our data obtained within the second Born approximation compares favorably with exact diagonalization results (available for up to 13 particles) and are expected to have predictive capability for substantially larger systems in the weak coupling limit.

Dynamics of Hubbard Nano-Clusters Following Strong Excitation

The Hubbard model is a prototype for strongly correlated electrons in condensed matter, for molecules and fermions or bosons in optical lattices. While the equilibrium properties of these systems have been studied in detail, the excitation and relaxation dynamics following a perturbation of the system are only poorly explored. Here, we present results for the dynamics of electrons following nonlinear strong excitation that are based on a nonequilibrium Green functions approach. We focus on small systems—“Hubbard nano-clusters”—that contain just a few particles where, in addition to the correlation effects, finite size effects and spatial inhomegeneity can be studied systematically. (© 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Ultrafast dynamics of finite Hubbard clusters: A stochastic mean-field approach

Finite lattice models are a prototype for strongly correlated quantum systems and capture essential properties of condensed matter systems. With the dramatic progress in ultracold atoms in optical lattices, finite fermionic Hubbard systems have become directly accessible in experiments, including their ultrafast dynamics far from equilibrium. Here, we present a theoretical approach that is able to treat these dynamics in any dimension and fully includes inhomogeneity effects. The method consists in stochastic sampling of mean-field trajectories and is found to be more accurate and efficient than current nonequilibrium Green functions approaches. This is demonstrated for Hubbard clusters with up to 512 particles in one, two and three dimensions.

Few-particle quantum dynamics–comparing nonequilibrium Green functions with the generalized Kadanoff–Baym ansatz to density operator theory

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].

Toward a Nonequilibrium Green Functions Approach to Diffusion in Strongly Coupled Finite Quantum Systems

Transport properties of strongly correlated quantum systems are of central interest in condensed matter, ultracold atoms and in dense plasmas. There, the proper treatment of strong correlations poses a great challenge to theory. Here, we apply a Nonequilibrium Green Functions approach using a lattice model as a basic system. This allow us to treat a finite spatially inhomogeneous system with an arbitrary nonequilibrium initial state. Placing all particles initially to one side of the system allows for a nonequilibrium study of diffusion. Strong correlation effects are incorporated via T-matrix selfenergies. (© 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Nonequilibrium Green function approach to the pair distribution function of quantum many-body systems out of equilibrium

The pair distribution function (PDF) is a key quantity for the analysis of correlation effects of a quantum system both in equilibrium and far from equilibrium. We derive an expression for the PDF in terms of the single-particle Green functions—the solutions of the Keldysh/Kadanoff-Baym equations in the two-time plane—for a one- or two-component system. The result includes initial correlations and generalizes previous density matrix expressions from single-time quantum kinetic theory. Explicit expressions for the PDF are obtained in second Born approximation.