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

The Kadanoff-Baym equations, which allow for the calculation of time-dependent expectation values of all one-particle observables, are found to yield unphysical electron density dynamics in the linear and nonlinear response, for $\mathrm{\Phi }$-derivable approximations, irrespective of interaction strength or type. In particular, we show that when calculated from the Kadanoff-Baym equations using correlated self-energy approximations, the linear response dynamics of isolated electron systems damps to an unphysical homogeneous density-distribution. The damping is also present for Hartree or Hartree-Fock self-energies. These surprising results supplement previous findings on the nonlinear response, and complement them by showing that the linear response is also plagued by unphysical dynamics. Being universal, this additional feature indicates the possible presence of an attractor that leads to amplitude death and a subsequent tendency to a homogeneous charge and density distribution. This unveils a scenario in which the Kadanoff-Baym dynamics simply breaks down, drastically restricting the parameter space for which the method can give physically meaningful insights. In addition to their relevance to the field of ultrafast electron dynamics in isolated and open systems, these findings may also impact the results obtained with the Bethe-Salpeter equation in linear response, due to the well-known equivalency between the two methods. This suggests the need for a different approach to the dynamics of quantum systems.

Figure 1

Long-time density dynamics at the perturbed site of Hubbard (solid line) and Coulomb (dotted line) dimers. Inset: The short-time density dynamics of the Hubbard dimer at the perturbed Site 1 (legend and axes in the notation of Ref. [19]).

Figure 2

Time-propagation up to 150 atomic units of time (a.u.t.): (a) Exact result for a Coulomb dimer $U=2$ with a constant field $E=0.05$. (b) Exact result for a Hubbard dimer $U=2$ and a kick of $E=0.05$ at $t=0$. (c), (d) Coulomb and Hubbard dimers, respectively, with a constant field of $E=0.05$ within the Second Born self-energy approximation ($U=1$ dotted line; $U=2$ dashed line; $U=3$ solid line). (e), (f) Coulomb and Hubbard dimers, respectively, with a kick of $E=0.05$ applied at $t=0$, within the Second Born self-energy approximation.

Figure 3

Time-dependent densities at Sites 1 and 2 of Coulomb and Hubbard dimers, with $U=2$ and constant field $E_{\text{tp}}$ applied to Site 1 during time-propagation (dashed-dotted lines). The same densities starting from a ground state with a constant shift $E_{\text{gs}}=0.05$ applied to Site 1, and no further time-dependent perturbation (solid lines).

Figure 4

(a) Time-dependent 2B densities at Site 1 of a symmetric and asymmetric four-site Hubbard dimer, with a constant $E_{\text{tp}}=0.05$ field applied during time-propagation. (b) Time-dependent $GW$ densities at Site 1 of a four-site Hubbard dimer, with constant external fields of different strengths applied to Site 1.

Figure 5

(a) Time-dependent Hartree-Fock densities of a Coulomb dimer $\left(U=5.0\right)$ starting from the ground state, and from a ground state with an applied field. Inset: Time-dependent Hartree densities of a Coulomb dimer $\left(U=2.5\right)$. (b) Time-dependent Hartree-Fock densities of a Hubbard dimer $\left(U=5.0\right)$ starting from a ground state and from a ground state with an applied field. Inset: Time-dependent Hartree densities of a Hubbard dimer $\left(U=2.5\right)$.

Figure 6

Time-dependent densities of a Coulomb dimer $\left(U=3.0\right)$. At $t=0$, a deltalike perturbation of $E=0.01$ was applied to Site 1. At $t=90$, a constant field of $E=0.05$ was switched on at the same site.