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Nonequilibrium Dynamics of the Hubbard Dimer

Y. Pavlyukh

Published Feb 14, 2024• 6 citations
Publisher

Abstract

Electron dynamics in a two‐sites Hubbard model is studied using the nonequilibrium Green's function approach. The study is motivated by the empirical observation that a full solution of the integro‐differential Kadanoff–Baym equation (KBE) is more stable and often accompanied by artificial damping [Marc Puig von Friesen, C. Verdozzi, and C.‐O. Almbladh (2009)] than its time‐linear reformulations relying on the generalized Kadanoff–Baym ansatz (GKBA). Additionally, for conserving theories, numerical simulations suggest that KBE produces natural occupations bounded by one and zero in agreement with the Pauli exclusion principle, whereas, in some regimes, GKBA‐based theories violate this principle. As the first step for understanding these issues, the electron dynamics arising in the adiabatic switching scenario is studied. Many‐body approximations are classified according to the channel of the Bethe–Salpeter equation in which electronic correlations are explicitly treated. They give rise to the so‐called second Born, T‐matrix, and GW approximations. In each of these cases, the model is reduced to a system of ordinary differential equations, which resemble equations of motion for a driven harmonic oscillator with time‐dependent frequencies. A more complete treatment of electronic correlations is achieved by combining different correlation channels, with parquet theory serving as a starting point.

DOI
10.1002/pssb.202300510

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Electron dynamics in a two‐sites Hubbard model is studied using the nonequilibrium Green's function approach. The study is motivated by the empirical observation that a full solution of the integro‐differential Kadanoff–Baym equation (KBE) is more stable and often accompanied by artificial damping [Marc Puig von Friesen, C. Verdozzi, and C.‐O. Almbladh (2009)] than its time‐linear reformulations relying on the generalized Kadanoff–Baym ansatz (GKBA). Additionally, for conserving theories, numerical simulations suggest that KBE produces natural occupations bounded by one and zero in agreement with the Pauli exclusion principle, whereas, in some regimes, GKBA‐based theories violate this principle. As the first step for understanding these issues, the electron dynamics arising in the adiabatic switching scenario is studied. Many‐body approximations are classified according to the channel of the Bethe–Salpeter equation in which electronic correlations are explicitly treated. They give rise to the so‐called second Born, T‐matrix, and GW approximations. In each of these cases, the model is reduced to a system of ordinary differential equations, which resemble equations of motion for a driven harmonic oscillator with time‐dependent frequencies. A more complete treatment of electronic correlations is achieved by combining different correlation channels, with parquet theory serving as a starting point.
Electron dynamics in a two-sites Hubbard model is studied using the nonequilibrium Green's function approach. The study is motivated by the empirical observation that a full solution of the integro-differential Kadanoff-Baym equation (KBE) is more stable and often accompanied by artificial damping [Marc Puig von Friesen, C. Verdozzi, and C.-O. Almbladh (2009)] than its time-linear reformulations relying on the generalized Kadanoff-Baym ansatz (GKBA). Additionally, for conserving theories, numerical simulations suggest that KBE produces natural occupations bounded by one and zero in agreement with the Pauli exclusion principle, whereas, in some regimes, GKBA-based theories violate this principle. As the first step for understanding these issues, the electron dynamics arising in the adiabatic switching scenario is studied. Many-body approximations are classified according to the channel of the Bethe-Salpeter equation in which electronic correlations are explicitly treated. They give rise to the so-called second Born, TTT-matrix and GWGWGW approximations. In each of these cases, the model is reduced to a system of ordinary differential equations, which resemble equations of motion for a driven harmonic oscillator with time-dependent frequencies. A more complete treatment of electronic correlations is achieved by combining different correlation channels, with parquet theory serving as a starting point.
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.
ABSTRACT Predicting real‐time dynamics in correlated systems is demanding: exact two‐time Green's function methods are accurate but often too costly, while the Generalized Kadanoff‐Baym Ansatz (GKBA) offers time‐linear propagation at the risk of uncontrolled behavior. We examine when and why GKBA fails in a minimal yet informative setting, the Holstein dimer that describes electron–phonon coupling. Using a conserving, fully self‐consistent electron–phonon self‐energy, we map out parameter regions where GKBA dynamics is stable and where it becomes unstable. We trace the onset of these failures to qualitative changes in the model's ground‐state solutions obtained from the full nonequilibrium Green's function theory, thereby providing practical stability bounds for GKBA time evolution. We further show that coupling the dimer to electronic leads can damp and, in part, cure these instabilities. The results supply simple diagnostics and guidelines for reliable GKBA simulations of electron–phonon dynamics.
Predicting real-time dynamics in correlated systems is demanding: exact two-time Green's function methods are accurate but often too costly, while the Generalized Kadanoff-Baym Ansatz (GKBA) offers time-linear propagation at the risk of uncontrolled behavior. We examine when and why GKBA fails in a minimal yet informative setting, the Holstein dimer that describes electron–phonon coupling. Using a conserving, fully self-consistent electron–phonon self-energy, we map out parameter regions where GKBA dynamics is stable and where it becomes unstable. We trace the onset of these failures to qualitative changes in the model's ground-state solutions obtained from the full nonequilibrium Green's function theory, thereby providing practical stability bounds for GKBA time evolution. We further show that coupling the dimer to electronic leads can damp and, in part, cure these instabilities. The results supply simple diagnostics and guidelines for reliable GKBA simulations of electron–phonon dynamics.
Abstract We apply a computationally efficient approach to study the time- and energy-resolved spectral properties of a two-site Hubbard model using the nonequilibrium Green’s function formalism. By employing the iterative generalized Kadanoff-Baym ansatz ( i GKBA) within a time-linear framework, we avoid the computational cost of solving the full two-time Kadanoff-Baym equations. Spectral information is extracted by coupling the system to multiple narrow-band leads, establishing a direct analogy to photoemission experiments. Our results reveal correlation-induced shifts and broadening of spectral features, along with a suppression of transient current oscillations. This approach provides a promising avenue for analyzing correlated electron dynamics in open quantum systems.
We present in detail a method we recently introduced (PRL. 103, 176404 (2009)) to describe finite systems in and out of equilibrium, where the evolution in time is performed via the Kadanoff-Baym Equations (KBE) within Many-Body Perturbation Theory (MBPT). The main property we analyze is the time-dependent density. We also study is the exchange-correlation potential of TDDFT, obtained via reverse engineering from the time-dependent density. Our systems consist of small, strongly correlated clusters, described by a Hubbard Hamiltonian within the Hartree-Fock, second Born, GW and T-matrix approximations. We compare the results from the KBE dynamics to those from exact numerical solutions. The outcome of our comparisons is that, among the many-body schemes considered, the T-matrix approximation is overall superior at all electron densities. Such comparisons permit a general assessment of the whole idea of applying MBPT, in the KBE sense, to finite systems. A striking outcome of our analysis is that when the system evolves under a strong external field, the KBE develop a steady-state solution as a consequence of a correlation-induced damping. This damping is present both in isolated (finite) systems, where it is purely artificial, as well as in clusters contacted to (infinite) macroscopic leads. To illustrate this point we present selected results for a system coupled to contacts within the T-matrix and second Born approximation. The extensive characterization we performed indicates that this behavior is present whenever approximate self energies, based upon infinite partial summations, are used. A second important result is that, for isolated clusters, the steady state reached is not unique but depends on how one switches on the external field. This may also true for clusters connected to leads.