Stability bounds for the generalized Kadanoff-Baym ansatz in the Holstein dimer

TL;DR

This work examines the dynamical stability of the Generalized Kadanoff-Baym Ansatz (GKBA) for electron-phonon dynamics in a minimal Holstein dimer using a conserving, self-consistent electron-phonon self-energy. It maps regions of stable and unstable GKBA time evolution and shows that instability onset coincides with ground-state bifurcations found in full nonequilibrium Green's function theory. By coupling the dimer to electronic leads within the wide-band limit, the authors demonstrate environment-induced damping that can regularize GKBA dynamics, providing practical stability bounds. The results offer concrete diagnostics and guidelines for reliable GKBA simulations and motivate extensions to larger lattices and structured reservoirs.

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.

Figures (5)

• Figure 1: Schematic of the Holstein dimer coupled to leads. The relative phonon mode—characterized by coupling $g$ and frequency $\omega_0$—is shown. $\Gamma_{\alpha}$ ($\alpha=L,R$) denotes the tunneling rate to the leads, and $t_0$ is the hopping term between sites.
• Figure 2: Total energy as a function of the effective interaction for different switching times. The region in panel (b) where the curves show discontinuities—blank gaps—corresponds to instabilities. The difference between the upper and lower critical values is indicated by $\Delta \lambda_c$ for the case $t_i=-80$.
• Figure 3: Unstable range of the effective interaction as a function of the adiabatic ratio for several switching times. Instability is signaled by the curves deviating from zero.
• Figure 4: Electronic natural occupation number and phononic occupation number as a function of time for different switching times. These temporal evolutions correspond to those of higher adiabatic ratios $\gamma$ in Fig. \ref{['fig:totalenergy']}b.
• Figure 5: Electronic natural occupation number and phononic occupation number as a function of time for the system connected to leads. Results are shown for several switching times.