Kadanoff-Baym dynamics of Hubbard clusters: Performance of many-body schemes, correlation-induced damping and multiple steady and quasi-steady states

Abstract

We present in detail a method we recently introduced ͓Phys. Rev. Lett. 103, 176404 ͑2009͔͒ to describe finite systems in and out of equilibrium, where the evolution in time is performed via the Kadanoff-Baym equations within many-body perturbation theory. 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 Kadanoff-Baym 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 superior at low electron densities while none of the tested approximations stands out at half filling. Such comparisons permit a general assessment of the whole idea of applying many-body perturbation theory, in the Kadanoff-Baym sense, to finite systems. A striking outcome of our analysis is that when the system evolves under a strong external field, the Kadanoff-Baym equations develop a steady-state solution as a consequence of a correlation-induced damping. This damping is present both in isolated ͑finite͒ systems, where it is purely artificial, as well as in clusters contacted to ͑infinite͒ macroscopic leads. The extensive numerical characterization we performed indicates that this behavior is present whenever approximate self-energies, which include correlation effects, are used. Another important result is that, for isolated clusters, the steady state reached is not unique but depends on how one switches on the external field. When the clusters are coupled to macroscopic leads, one may reach multiple quasisteady states with arbitrarily long lifetimes. .Ϫb the GW approximation ͑GWA͒, 21 and the T-matrix approximation ͑TMA͒. 22,23 All these approximations are conserving, 1 which clearly is of great importance when propagating the KBE, and all of them, apart from HFA, have self-energies which are nonlocal in space and time. For the GWA, we will consider both a spin-independent and a spindependent version. 11,24 The latter has the advantage to alleviate the effect of self-screening. 25,26

On the dangers of partial diagrammatic summations: Benchmarks for the two-dimensional Hubbard model in the weak-coupling regime

“…While some tests of the GW [24,25], TMA [25,26], or FLEX [27] approaches have been published, we still lack a clear picture about the importance of the different diagram classes, and the beneficial or detrimental effect of self-consistent partial resummations.…”

Some open questions in TDDFT: Clues from lattice models and Kadanoff–Baym dynamics

“…In this way non-locality in space and memory effects can be properly included, once v xc is retrieved from the many-body self-energy via the time-dependent Sham-Schlüter equation [90]. This well established relation between many-body perturbation theory and TDDFT on the Keldysh contour was numerically examined in [113], by looking at exchangecorrelation potentials obtained via time-dependent reverse engineering using the time-dependent densities from the KBE. The performance of two self-interaction correction schemes for the 1D Hubbard model has been scrutinized in [91], while a shortcoming of the BALDA was pointed out in [92], specifically its capability to reproduce correctly the Friedel oscillations in the inhomogeneous 1D Hubbard model.…”

“…Eq. (25) and the one corresponding to t 2 are solved numerically, and for non-isolated systems, Σ contains an additional contribution, the embedding self-energy Σ emb , to treat the coupling system-environment [103,109,113].…”

“…In this case the self-energy is Σ(12) = G(12)W (12). In a spin-dependent treatment of the TMA [113,116] All these approximations are conserving [40], i.e. the macroscopic conservation laws (for the energy, the number of particles, etc.)…”

Partial self-consistency and analyticity in many-body perturbation theory: Particle number conservation and a generalized sum rule

“…In fully self-consistent calculations, it has been shown [12,13] that strong time-dependent fields can yield artificial steady states in finite systems. Partial self-consistency lessened this effect.…”