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.
The reduced density matrix of an interacting system can be used as the basis for a truncation scheme, or in an unbiased method to discover the strongest kind of correlation in the ground state. In this paper, we investigate the structure of the many-body fermion density matrix of a small cluster in a square lattice. The cluster density matrix is evaluated numerically over a set of finite systems, subject to nonsquare periodic boundary conditions given by the lattice vectors ${\mathbf{R}}_{1}\ensuremath{\equiv}({R}_{1x},{R}_{1y})$ and ${\mathbf{R}}_{2}\ensuremath{\equiv}({R}_{2x},{R}_{2y})$. We then approximate the infinite-system cluster density-matrix spectrum by averaging the finite-system cluster density matrix (i) over degeneracies in the ground state, and orientations of the system relative to the cluster, to ensure it has the proper point-group symmetry; and (ii) over various twist boundary conditions to reduce finite size effects. We then compare the eigenvalue structure of the averaged cluster density matrix for noninteracting and strongly interacting spinless fermions, as a function of the filling fraction
n, and discuss whether it can be approximated as being built up from a truncated set of single-particle operators.