Nonequilibrium Green Functions Simulations for Large Correlated Systems

Correlated real-time dynamics in large, spatially inhomogeneous quantum systems remain difficult to access with nonequilibrium many-body methods. Two-time nonequilibrium Green functions (NEGF) retain dynamical correlations but their computational runtime grows cubically with the number of time steps $N_\mathrm{t}$. This scaling bottleneck could recently be overcome by introducing the G1--G2 scheme that is linear in $N_\mathrm{t}$, but requires propagation of a two-particle correlation function and may suffer from numerical instabilities. This has restricted simulations to small systems with $N_\mathrm{b} \sim 10^2$ basis states. Here we introduce a quantum-fluctuation formulation of nonequilibrium Green functions, denoted $\delta$NEGF, that represents dynamical two-particle correlations through fluctuations of field-operator products, $\delta \hat G$. This guarantees stable dynamics by preserving the positivity of the reduced density matrices, avoids the explicit storage of the two-particle Green function, and reduces the propagation to a finite ensemble of Hartree-Fock-like trajectories. Combined with a stochastic low-rank decomposition of the correlation functions, the method retains time-linear scaling while extending dynamical $GW$ and particle-particle and particle-hole $T$-matrix simulations to basis sizes of order $N_\mathrm{b}\sim 10^4$. We benchmark $\delta$NEGF against exact and HF-GKBA results for lattice systems, finding stable correlated dynamics also at strong coupling. We further demonstrate large-scale simulations of diffusion in two-dimensional Hubbard lattices and ultrafast relaxation in graphene nanoribbon heterostructures with long-range Coulomb interactions. These results establish $\delta$NEGF as a scalable route to dynamical self-energy simulations of large, spatially inhomogeneous correlated quantum systems beyond the reach of existing NEGF implementations.