In a recent paper we presented a stochastic Green function (SGF) algorithm, which has the properties of being general and easy to apply to any lattice Hamiltonian of the form $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathcal{H}}=\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathcal{V}}\ensuremath{-}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathcal{T}}$, where $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathcal{V}}$ is diagonal in the chosen occupation number basis and $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathcal{T}}$ has only positive matrix elements. We propose here a modified version of the update scheme that keeps the simplicity and generality of the original SGF algorithm, and significantly enhances its efficiency.
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
Nt. This scaling bottleneck could recently be overcome by introducing the G1--G2 scheme that is linear in
Nt, but requires propagation of a two-particle correlation function and may suffer from numerical instabilities. This has restricted simulations to small systems with
Nb∼102 basis states. Here we introduce a quantum-fluctuation formulation of nonequilibrium Green functions, denoted
δNEGF, that represents dynamical two-particle correlations through fluctuations of field-operator products,
δG^. 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
Nb∼104. We benchmark
δNEGF 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
δNEGF as a scalable route to dynamical self-energy simulations of large, spatially inhomogeneous correlated quantum systems beyond the reach of existing NEGF implementations.
A recent technique, proposed to alleviate the ``sign problem disease,'' is discussed in detail. As is well known, the ground state of a given Hamiltonian H can be obtained by applying the propagator ${e}^{\ensuremath{-}H\ensuremath{\tau}}$ to a trial wave function ${\ensuremath{\psi}}_{T}$ and sampling statistically the state ${\ensuremath{\psi}}_{\ensuremath{\tau}}{=e}^{\ensuremath{-}H\ensuremath{\tau}}{\ensuremath{\psi}}_{T}$ for large imaginary time $\ensuremath{\tau}.$ However, the sign problem may appear in the simulation and such statistical propagation would be practically impossible without employing some approximation such as the ``fixed node'' (FN) one. The present method allows the improvement of the FN dynamics with a systematic correction scheme. This is possible by the simple requirement that, after a short imaginary time propagation via the FN Hamiltonian, a number p of correlation functions can be further constrained to be exact by small perturbations of the FN state, which is free from the sign problem. By iterating this procedure, the Monte Carlo average sign, which is almost zero when there is a sign problem, remains stable and finite even for large $\ensuremath{\tau}.$ The proposed algorithm is tested against exact diagonalization data available on finite lattices. It is also shown, in some test cases, that the dependence of the results upon the few parameters entering the stochastic technique can be very easily controlled, unless for exceptional cases.
We present a numerically exact continuous-time quantum Monte Carlo algorithm for fermions with a general interaction nonlocal in space-time. The new determinantal grand-canonical scheme is based on a stochastic series expansion for the partition function in the interaction representation. The method is particularly applicable for multiband, time-dependent correlations since it does not invoke the Hubbard-Stratonovich transformation. The test calculations for exactly solvable models, as well results for the Green function and for the time-dependent susceptibility of the multiband supersymmetric model with a spin-flip interaction are discussed.
We show how the worldline quantum Monte Carlo procedure, which usually relies on an artificial time discretization, can be formulated directly in continuous time, rendering the scheme exact. For an arbitrary system with discrete Hilbert space, none of the configuration update procedures contain small parameters. We find that the most effective update strategy involves the motion of worldline discontinuities (both in space and time), i.e., the evaluation of the Green’s function. Being based on local updates only, our method nevertheless allows one to work with the grand canonical ensemble and nonzero winding numbers, and to calculate any dynamical correlation function as easily as expectation values of, e.g., total energy. The principles found for the update in continuous time generalize to any continuous variables in the space of discrete virtual transitions, and in principle also make it possible to simulate continuous systems exactly.