Dynamically screened ladder approximation: Simultaneous treatment of strong electronic correlations and dynamical screening out of equilibrium

Dynamical screening is a key property of charged many-particle systems. Its theoretical description is based on the $GW$ approximation that is extensively applied for ground-state and equilibrium situations but also for systems driven out of equilibrium. The main limitation of the $GW$ approximation is the neglect of strong electronic correlation effects that are important in many materials as well as in dense plasmas. Here we derive the dynamically screened ladder (DSL) approximation that self-consistently includes, in addition to the $GW$ diagrams, also particle-particle and particle-hole $T$-matrix diagrams. The derivation is based on reduced-density-operator theory and the result is equivalent to the recently presented G1-G2 scheme [N. Schlünzen et al., Phys. Rev. Lett. 124, 076601 (2020); J.-P. Joost et al., Phys. Rev. B 101, 245101 (2020)]. We perform extensive time-dependent DSL simulations for finite Hubbard clusters and present tests against exact results that confirm excellent accuracy as well as total-energy conservation of the approximation. At strong coupling and for long simulation durations, instabilities are observed. These problems are solved by enforcing contraction consistency and applying a purification approach.

Figure 1

Keldysh “round-trip” time contour that is used in NEGF theory to treat initial correlations via “adiabatic switching” of the pair interaction, starting from an uncorrelated state in the remote past (for more details, see Refs. [11, 63, 64, 65]).

Figure 2

Self-energy diagrams for the perturbative (with respect to powers of the interaction) approach up to order three. Hartree-Fock (HF) contains contributions of first order in $w$, the second-order self-energy (SOA) contains second-order diagrams (together with the first order), whereas the third-order approximation (TOA) contains all diagrams up to order $w^3$. Note that the second diagrams of order $w^1$ and $w^2$ describe exchange processes, respectively. For order $w^3$, exchange processes are included in the latter six diagrams.

Figure 3

Self-energy diagrams for the three resummation approaches starting from the second-order contributions. Dots indicate continuation of the sums to infinite order. Note that for TPP it is possible to also include the corresponding exchange diagrams (second line).

Figure 4

Illustration of the instability of the G1-G2 scheme for a half-filled six-site Hubbard system at moderate coupling $U/J=4$. Sites 1–3 are initially doubly occupied and sites 4–6 are empty. At time $t=0$ the confinement potential is removed (confinement quench). In contrast to the exact (full black line) and DSL* solutions (dashed-dotted yellow line), the TOA (dashed green line) and DSL solutions (dashed-dotted red line) become unstable already at time $tJ/\hslash \approx 4$ and $tJ/\hslash \approx 6$, respectively. (a) Occupation $n$ of site 1. (b), (c) Correlation and kinetic energy, respectively. (d) Largest eigenvalue of the two-particle Green function.

Figure 5

Evolution of the interaction energy following an interaction quench. The ideal ground state is prepared for a six-site Hubbard chain after which the interaction is instantaneously switched from $U=0$ to $U=J$ at $t=0$. (a) Corresponds to a filling of $n=\frac{1}{6}$ and (b) to $n=\frac{1}{3}$. Exact results are shown in black. The green (orange) curves correspond to the TOA (DSL*) results.

Figure 6

Density dynamics on the first site of a six-site Hubbard system following a sudden switch on of an external potential, for (a) $U=1$ and (b) $U=2$. The dynamics start from the interacting ground state (generated by the adiabatic switching) and, at $t=0$, a constant local potential of $w_0=J$ is applied to the first site. Exact results are shown in black. The green (orange) curves correspond to the TOA (DSL*) results.

Figure 7

Same setup as in Fig. 4 but for various interactions from $U=J$ to $U=4J$. (a), (b) Correlation and kinetic energy of the system, respectively. Each curve beyond $U=J$ is shifted vertically by $2J$ with respect to the previous one. The TOA results (dashed green line) become increasingly unstable (diverge earlier) for increasing interaction strength.

Figure 8

Relaxation of a CDW state of doublons. Shown is the interaction energy for $U/J=1,2,3$ and $L=N=20$. The present DSL results with contraction consistency and purifciation (DSL*, orange) are compared to the DMRG benchmark (black) and to G1-G2–TOA (green) simulations.