Comment on “On the unphysical solutions of the Kadanoff-Baym equations in linear response: Correlation-induced homogeneous density-distribution and attractors”

In a recent Rapid Communication [A. Stan, Phys. Rev. B 93, 041103(R) (2016)], the reliability of the Keldysh-Kadanoff-Baym equations (KBE) using correlated self-energy approximations applied to linear and nonlinear response has been questioned. In particular, the existence of a universal attractor has been predicted that would drive the dynamics of any correlated system towards an unphysical homogeneous density distribution regardless of the system type, the interaction, and the many-body approximation. Moreover, it was conjectured that even the mean-field dynamics would be damped. Here, by performing accurate solutions of the KBE for situations studied in that paper, we prove these claims wrong, being caused by numerical inaccuracies.

Figure 1

Density evolution on the first Hubbard site of the dimer with $U=4$ after the switch-on of a constant excitation with $w_0=5$ at site 1, which is shown in the lower left inset. The insets on the right-hand-side show the asymptotic density distributions of Ref. [23] (top) and the present work (bottom). Solutions of the KBE in self-consistent second Born approximation. The time step in our simulation is $\mathrm{\Delta }t={10}^{-3}$.

Figure 2

Density evolution at site 1 of a Hubbard dimer ($U=3$), following a very weak ($w_0=0.05$) steplike excitation at site 1. Black dashed line: result of Ref. [23]. Full red line: present result, using a time step of $\mathrm{\Delta }t={10}^{-3}$.

Figure 3

Mean field (Hartree) density evolution of a Hubbard dimer with $U=5$ following the switch-on of a constant excitation with $w_0=0.01$ on site 1. The results of Ref. [23] are shown by the red and black lines and exhibit damping, whereas our results are undamped (orange and brown lines). The high-frequency oscillations of the density are illustrated in the inset (the resolution of the results of Ref. [23] does not allow to resolve these oscillations).

Figure 4

Demonstration of the convergence behavior for a Hubbard dimer ($U=3$), following a very weak ($w_0=0.05$) steplike excitation at site 1. (a)/(b): density evolution at site 1 for different time steps $\mathrm{\Delta }t$. Simulations in (a) [(b)] are done with method I (method II). (c) Quality of energy conservation corresponding to the results in (a). (d). Different energy contributions for $\mathrm{\Delta }t=3\times {10}^{-1}$ in (a) and (c). We note that in (b) the time steps only refer to the integration (B). The collision integral (A) is solved with $\mathrm{\Delta }t={10}^{-2}$.

Figure 5

Illustration of the trapezoidal rule for a typical calculation of the collision integral. The red line shows a realistic example of the integrand in Eq. (1) of Ref. [23] during a converged simulation. The blue curve corresponds to the respective approximation of the trapezoidal rule for a large time step $\mathrm{\Delta }t=0.3$.

Figure 6

Time reversal properties of the density on the first site for a Hubbard dimer with steplike excitation. Solid yellow (dashed brown) lines correspond to simulations where $\widehat{H}\left(t\right)\to -\widehat{H}(-t)$ ($t\to -t$) is being applied. All calculations are performed via method I. Parts (a) and (b) show the density for linear response ($w_0=0.05$) and $U=3$. The time step in (a) is $\mathrm{\Delta }t=3\times {10}^{-1}$ (not converged), while in (b) it is $\mathrm{\Delta }t={10}^{-2}$ (converged). Part (c) shows the time reversal behavior for a strong excitation ($w_0=5$) with $U=4$ and a time step of $\mathrm{\Delta }t=2.5\times {10}^{-3}$ (converged).