Quantum fluctuations approach to the nonequilibrium $GW$ approximation: Density correlations and dynamic structure factor

The quantum dynamics of correlated fermionic or bosonic many-body systems following external excitation can be successfully studied using nonequilibrium Green functions (NEGFs) or reduced density matrix methods. Approximations are introduced via a proper choice of the many-particle self-energy or decoupling of the BBGKY-hierarchy, respectively. These approximations are based on Feynman's diagram approaches or on cluster expansions into single-particle and correlation operators. In a recent paper [E. Schroedter, J.-P. Joost, and M. Bonitz, Condens. Matter Phys. 25, 23401 (2022)] we presented a different approach in which, instead of equations of motion for the many-particle NEGF (or density operators), equations for the correlation functions of fluctuations are analyzed. In particular, we derived the stochastic $GW$ and polarization approximations that are closely related to the nonequilibrium $GW$ approximation. Here, we extend this approach to the computation of two-time observables depending on the specific ordering of the underlying operators. In particular, we apply this extension to the calculation of the density correlation function and dynamic structure factor of correlated Hubbard clusters in and out of equilibrium.

Figure 1

Illustration of the steps to go from the Keldysh-Kadanoff-Baym equations (KBEs) using the $GW$ approximation (with exchange contributions) to the time-local stochastic fluctuations approach using the SGW approximation (SPA). For details, see the text.

Figure 2

Connection between three concepts: the KBE with Hartree self-energy and an external field, the BSE for the XC function [Eq. (36)] with two-particle irreducible vertex (38), and the present two-time quantum fluctuations approach [Eqs. (30) and (31)].

Figure 3

Fourier transform of the density response function (arbitrary units) for the first site of a half-filled Hubbard dimer, without PBC, using SPA-ME with (blue lines) and without (orange lines) a kick excitation at $U=0.0J$ (a) and $U=1.0J$ (b). The results are compared to the Fourier transform of the analytical result. The damping constant was chosen to be $\eta =0.005J/\hslash$.

Figure 4

Coupling dependence of the excitation energy of a Hubbard dimer for the dipole-allowed transition from the ground state, cf. Fig. 3. Comparison of the present SPA-ME simulations with (blue line) and without (orange line) a kick excitation to the analytical solution.

Figure 5

Fourier transform of the density response function (arbitrary units) for the first site of a half-filled six-site system using SPA-ME with (blue lines) and without (orange lines) kick excitation at $U=0.0J$ (a) and $U=1.0J$ (b). The results are compared to an exact diagonalization (CI) calculation where a kick excitation, Eq. (78), was used. A damping constant of $\eta =0.03J/\hslash$ for the exponential damping of the time propagation was used.

Figure 6

(a) Fourier transform of the density response function (arbitrary units) for the first site of a half-filled 50-site ring at $U=0.0J$ using SPA-ME. The results are compared to the analytical result. (b) Relative error of the Fourier-transformed density response. A damping constant of $\eta =0.02J/\hslash$ was used.

Figure 7

Fourier transform of the density response function (arbitrary units) of a half-filled 50-site ring at $U=0.0J$ (left) and $U=1.0J$ (right) using SPA-ME. The dashed lines denote the corresponding peak positions from RPA calculations. A damping constant of $\eta =0.2J\hslash$ was used.

Figure 8

Dynamic structure factor of a half-filled 50-site ring at $q=\pi /a_0$ for different $U$ using SPA-ME (solid lines) and RPA (dashed lines).

Figure 9

Dynamic structure factor of a half-filled 50-site ring at $q_1=3\pi /\left(5a_0\right)$ (solid lines) and $q_2=\pi /\left(5a_0\right)$ (dash-dotted lines) for different $U$ using SPA-ME.

Figure 10

Dynamic structure factor for a half-filled 50-site ring at $U=1.0J$ at $q=\pi /a_0$ from SPA-ME with (blue line) and without (orange line) a kick excitation and RPA. The time propagated results were exponentially damped with damping constant $\eta =0.2J/\hslash$.

Figure 11

Time-dependent density perturbation at the first site for a half-filled six-site chain for three coupling strengths [$U=0.0J$ (a), $U=0.5J$ (b), and $U=1.0J$ (c)] following a kick excitation using SPA (blue lines) and CI. The data are compared to linear-response results from a ground-state SPA-ME calculation (orange lines); cf. Eq. (94).

Figure 12

Density response function for a half-filled ring with six sites at $U=0.1J$ following a confinement quench at $t=0$. Upper left (lower right) triangle corresponds to SPA-ME (CI) results.

Figure 13

Same as Fig. 12, but at $U=0.4J$.

Figure 14

Density response function for different times $T$ following a confinement quench, for a half-filled ring with six sites at $U=0.1J$. Comparison of the present SPA-ME approach to CI calculations.

Figure 15

Same as Fig. 14, but at $U=0.4J$.

Figure 16

Density evolution at site 1 of a half-filled ring with six sites following a confinement quench.

Figure 17

Density response function for different center-of-mass times $T$ following a confinement quench, for a half-filled ring with 30 sites at $U=0.1J$ and $0.4J$ using SPA-ME.

Figure 18

Fourier transform of the density response function with respect to the relative time for a half-filled ring with 30 sites at fixed center-of-mass times, $T$, at $U=0.1J$ and $0.4J$ using SPA-ME. The $U=0.1J$ results in (a), (b), and (c) have been multiplied with a factor of 5. A damping constant of $\eta =0.04J/\hslash$ was used.

Figure 19

Same as Fig. 18, but for the ground state at $U=0.1J$ and $0.4J$.