Accelerating nonequilibrium Green function simulations with embedding self-energies

Real-time nonequilibrium Green functions (NEGFs) have been very successfully used to simulate the dynamics of correlated many-particle systems far from equilibrium. However, NEGF simulations are computationally expensive since the effort scales cubically with the simulation duration. Recently, we introduced the G1–G2 scheme that allows for a dramatic reduction to time-linear scaling [N. Schlünzen et al., Phys. Rev. Lett. 124, 076601 (2020); J.-P. Joost et al., Phys. Rev. B 101, 245101 (2020)]. Here we tackle another problem: the rapid growth of the computational effort with the system size. In many situations where the system of interest is coupled to a bath, to electric contacts, or to similar macroscopic systems for which a microscopic resolution of the electronic properties is not necessary, efficient simplifications are possible. This is achieved by the introduction of an embedding self-energy—a concept that has been successful in standard NEGF simulations. Here, we demonstrate how the embedding concept can be introduced into the G1–G2 scheme, allowing us to drastically accelerate NEGF embedding simulations. The approach is compatible with all advanced self-energies that can be represented by the G1–G2 scheme [as described in J.-P. Joost et al., Phys. Rev. B 105, 165155 (2022)] and retains the memoryless structure of the equations and their time-linear scaling. As a numerical illustration, we investigate the charge transfer between a Hubbard nanocluster and an additional site which is of relevance for the neutralization of ions in matter.

Figure 1

Strong charge exchange between a six-site Hubbard chain and a single site with index “0.” Shown are the time-dependent electron densities on the three sites “1,” “2,” and “3” of the chain that are adjacent to the external site. Initially, the chain is at half filling. (a),(b) The lattice electrons are noninteracting ($U=0$) and the additional site is empty, $n_0^e=0$. (c) Interaction of the electrons is treated on the Hartree-Fock level with $U=4J$ and $n_0^e=0.3$. The black solid lines indicate the function $\gamma \left(t\right)$ (scaled by a factor $1/3$) with the pulse width ${\tau }_{\gamma }=1t_0$ in all panels. The amplitude equals (a) ${\gamma }_0=0.5J$ and (b),(c) ${\gamma }_0=3J$. Three sets of results are shown: two-time NEGF embedding results (full lines), the G1–G2 model of Sec. 3 (dots), and the extended embedding model, given by Eqs. (40) and (41) (dashes).

Figure 2

(a) Density of states of the 50-site Hubbard chain and four cases of the position of the energy $\epsilon$ of the additional site (for better visibility, all discrete states were Gaussian broadened). (b),(c) Time evolution of the density on the attached site “0” and on the first site “1” of the chain, respectively. Parameters: ${\gamma }_0=2J, n_0^{\text{e}}=0.3$, and ${\tau }_{\gamma }=1.311t_0$. The line styles distinguish different Hubbard interaction strengths $U$, given in the inset of (c).

Figure 3

Total charge transfer $\mathrm{\Delta }{N}^{s\to e}$ as a function of the pulse length ${\tau }_{\gamma }$ for the system of Fig. 2: (a) ${\gamma }_0=0.5J$, (b) ${\gamma }_0=2J$, and (c) ${\gamma }_0=4J$. In the numerical simulations, we used $t_{\gamma }=50t_0$ and extracted the value of $\mathrm{\Delta }{N}^{s\to e}$ at time $t=150t_0$. For very large ${\gamma }_0$, reflections at the other end of the chain influence the adiabatic results around ${\tau }_{\gamma }\sim 10t_0$; cf. the red lines in (c). Furthermore, the thin black curves are functions of the form $f\left({\tau }_{\gamma }\right)=aexp(-b{\tau }_{\gamma })$ and denote fits to the tails of the blue solid lines for ${\tau }_{\gamma }\to 10t_0$.