The density of vibrational states of harmonic chains is calculated with the Green function formalism. Chains with impurities are considered, and it is explicitly shown how the localized states are built up from the contribution of every lattice site to the density of states. The method can be used, with slight changes, to study the dynamics of a linear ferromagnetic lattice with nearest-neighbor interactions. The procedure employed in this article shows clearly the power and simplicity of the Green function method.The density of vibrational states of harmonic chains is calculated with the Green function formalism. Chains with impurities are considered, and it is explicitly shown how the localized states are built up from the contribution of every lattice site to the density of states. The method can be used, with slight changes, to study the dynamics of a linear ferromagnetic lattice with nearest-neighbor interactions. The procedure employed in this article shows clearly the power and simplicity of the Green function method.

American Journal of PhysicsFeb 1, 19832 citations

Nonequilibrium Green Functions Simulations for Large Correlated Systems

Erik Schroedter, M. Bönitz, Jan‐Philip Joost

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 $δ$ NEGF, that represents dynamical two-particle correlations through fluctuations of field-operator products, $δ\hat G$ . 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 $δ$ 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 $δ$ NEGF as a scalable route to dynamical self-energy simulations of large, spatially inhomogeneous correlated quantum systems beyond the reach of existing NEGF implementations.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 $δ$NEGF, that represents dynamical two-particle correlations through fluctuations of field-operator products, $δ\hat G$. 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 $δ$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 $δ$NEGF as a scalable route to dynamical self-energy simulations of large, spatially inhomogeneous correlated quantum systems beyond the reach of existing NEGF implementations.

arXiv:2606.10773Jun 9, 20260 citations

The Green Function Method in Statistical Mechanics

D. ter Haar

Elsevier eBooksJan 1, 19774 citations

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

δ

NEGF, that represents dynamical two-particle correlations through fluctuations of field-operator products,

δ\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

δ

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

δ

NEGF as a scalable route to dynamical self-energy simulations of large, spatially inhomogeneous correlated quantum systems beyond the reach of existing NEGF implementations.