We extend the stochastic GW (sGW) formalism to fully spin-polarized systems, encompassing both collinear and non-collinear spin configurations. For non-collinear systems-where Kohn-Sham states are complex two-component spinors-we develop a complex-valued stochastic basis that preserves the real-valued external stochastic charge applied at time zero. This basis enables an unbiased evaluation of the random-phase approximation (RPA) screened interaction for spinors. Through error analysis and tests on real materials, we show that the performance of collinear sGW retains the same time complexity as the spin-unpolarized sGW . The non-collinear sGW incurs a computational cost two to three times higher than the spin-unpolarized version, while preserving linear scaling with low multiplicity. By unifying collinear and non-collinear treatments within a single scalable framework, our work paves the way for routine many-body predictions in large scale magnetic and spin-orbit-coupled material systems.
We extend the stochastic GW (sGW) formalism to fully spin-polarized systems, encompassing both collinear and non-collinear spin configurations. For non-collinear systems-where Kohn-Sham states are complex two-component spinors-we develop a complex-valued stochastic basis that preserves the real-valued external stochastic charge applied at time zero. This basis enables an unbiased evaluation of the random-phase approximation (RPA) screened interaction for spinors. Through error analysis and tests on real materials, we show that the performance of collinear sGW retains the same time complexity as the spin-unpolarized sGW . The non-collinear sGW incurs a computational cost two to three times higher than the spin-unpolarized version, while preserving linear scaling with low multiplicity. By unifying collinear and non-collinear treatments within a single scalable framework, our work paves the way for routine many-body predictions in large scale magnetic and spin-orbit-coupled material systems.
We extend the stochastic GW (s GW ) formalism to fully spin-polarized systems, encompassing both collinear and noncollinear spin configurations. For noncollinear systems, where Kohn–Sham states are complex two-component spinors, we develop a complex-valued stochastic basis that preserves the real-valued external stochastic charge applied at time zero. This basis enables an unbiased evaluation of the random-phase approximation (RPA) screened interaction for spinors. Through error analysis and tests on real materials, we show that the performance of collinear s GW retains the same time complexity as the spin-unpolarized s GW . The noncollinear s GW incurs a computational cost two to three times higher than the spin-unpolarized version, while preserving linear scaling with low multiplicity. By unifying collinear and noncollinear treatments within a single scalable framework, our work paves the way for routine many-body predictions in large scale magnetic and spin–orbit–coupled material systems.
We develop a formalism to calculate the quasiparticle energy within the GW many-body perturbation correction to the density functional theory. The occupied and virtual orbitals of the Kohn-Sham Hamiltonian are replaced by stochastic orbitals used to evaluate the Green function G, the polarization potential W, and, thereby, the GW self-energy. The stochastic GW (sGW) formalism relies on novel theoretical concepts such as stochastic time-dependent Hartree propagation, stochastic matrix compression, and spatial or temporal stochastic decoupling techniques. Beyond the theoretical interest, the formalism enables linear scaling GW calculations breaking the theoretical scaling limit for GW as well as circumventing the need for energy cutoff approximations. We illustrate the method for silicon nanocrystals of varying sizes with N_{e}>3000 electrons.
The quantum dynamics of fermionic or bosonic many-body systems following external excitation can be successfully studied using two-time nonequilibrium Green's functions (NEGF) or single-time reduced density matrix methods. Approximations are introduced via a proper choice of the many-particle self-energy or decoupling of the BBGKY hierarchy. These approximations are based on Feynman's diagram approaches or on cluster expansions into single-particle and correlation operators. Here, we develop a different approach where, instead of equations of motion for the many-particle NEGF (or density operators), single-time equations for the correlation functions of fluctuations are analyzed. We present a derivation of the first two equations of the alternative hierarchy of fluctuations and discuss possible decoupling approximations. In particular, we derive the polarization approximation (PA) which is shown to be equivalent to the single-time version [following by applying the generalized Kadanoff-Baym ansatz (GKBA)] of the nonequilibrium GW approximation with exchange effects of NEGF theory, for weak coupling. The main advantage of the quantum fluctuations approach is that the standard ensemble average can be replaced by a semiclassical average over different initial realizations, as was demonstrated before by Lacroix and co-workers [see e.g. D. Lacroix et al., Phys. Rev. B, 2014, 90, 125112]. Here, we introduce the stochastic GW (SGW) approximation and the stochastic polarization approximation (SPA) which are demonstrated to be equivalent to the single-time GW approximation without and with exchange, respectively, in the weak coupling limit. In addition to the standard stochastic approach to sample initial configurations, we also present a deterministic approach. Our numerical tests confirm that our approach has the same favorable linear scaling with the computation time as the recently developed G1-G2 scheme [Schluenzen et al., Phys. Rev. Lett., 2020, 124, 076601]. At the same time, the SPA and SGW approaches scale more favorably with the system size than the G1-G2 scheme, allowing to extend nonequilibrium GW calculations to bigger systems.
We present a many-body GW formalism for quantum subsystems embedded in discrete polarizable environments containing up to several hundred thousand atoms described at a fully ab initio random phase approximation level. Our approach is based on a fragment approximation in the construction of the Green's function and independent-electron susceptibilities. Further, the environing fragments susceptibility matrices are reduced to a minimal but accurate representation preserving low order polarizability tensors through a constrained minimization scheme. This approach dramatically reduces the cost associated with inverting the Dyson equation for the screened Coulomb potential W, while preserving the description of short to long-range screening effects. The efficiency and accuracy of the present scheme is exemplified in the paradigmatic cases of fullerene bulk, surface, subsurface, and slabs with varying number of layers.
Based on the random phase approximation (RPA), RPA renormalization [J. E. Bates and F. Furche, J. Chem. Phys. 139, 171103 (2013)] is a robust many-body perturbation theory that works for molecules and materials because it does not diverge as the Kohn-Sham gap approaches zero. Additionally, RPA renormalization enables the simultaneous calculation of RPA and beyond-RPA correlation energies since the total correlation energy is the sum of a series of independent contributions. The first-order approximation (RPAr1) yields the dominant beyond-RPA contribution to the correlation energy for a given exchange-correlation kernel, but systematically underestimates the total beyond-RPA correction. For both the homogeneous electron gas model and real systems, we demonstrate numerically that RPA renormalization beyond first order converges monotonically to the infinite-order beyond-RPA correlation energy for several model exchange-correlation kernels and that the rate of convergence is principally determined by the choice of the kernel and spin polarization of the ground state. The monotonic convergence is rationalized from an analysis of the RPA renormalized correlation energy corrections, assuming the exchange-correlation kernel and response functions satisfy some reasonable conditions. For spin-unpolarized atoms, molecules, and bulk solids, we find that RPA renormalization is typically converged to 1 meV error or less by fourth order regardless of the band gap or dimensionality. Most spin-polarized systems converge at a slightly slower rate, with errors on the order of 10 meV at fourth order and typically requiring up to sixth order to reach 1 meV error or less. Slowest to converge, however, open-shell atoms present the most challenging case and require many higher orders to converge.
Methods based on the many-body Green's function are generally accepted as the path forward beyond Kohn-Sham based density functional theory, in order to compute from first principles electronic structure of materials with strong correlations and excited-state properties in nano- and materials science. Here we present an efficient method to compute the screened Coulomb interactionW, the crucial and computationally most demanding ingredient in the GW method, within the framework of the all-electron Linearized Augmented Plane Wave method. We use the method to compute from first principles, within the constrained random phase approximation (c-RPA), the frequency-dependent screened Hubbard U-matrix defined for a Wannier basis in which we downfold the many-body Hamiltonian for La2CuO4, the canonical parent compound of several cuprate high-temperature superconductors. These results were computed at scale on the Cray XT5 at ORNL, sustaining 1.30 petaflop. We discuss the details of the algorithm and its implementation that allowed us to reach high efficiency and short time to solution on today's petaflop supercomputers.