Session G46: Excited State I: Method development: Many-Body Perturbation Theory

Massively parallel cubic-scaling GW calculations with the OpenAtom software

OpenAtom is an open source, massively parallel ab initio density functional software package based on plane waves and pseudopotentials (http://charm.cs.uiuc.edu/OpenAtom) that takes advantage of the Charm++ parallel framework.  We will describe a cubic-scaling massively parallel implementation of the GW approximation (L. Hedin, Phys. Rev. 139, 1965) to the electronic self-energy that is useful for beyond-DFT calculations of electronic band structures.   After a brief summary of the OpenAtom and Charm++ frameworks along with the main ideas behind our cubic-scaling approach (Kim, Martyna, Ismail-Beigi, Phys. Rev. B 101, 2020), we will describe the approach we employ for its parallelization (along with benefits and tradeoffs involved) and some examples of its parallel performance on large supercomputers.   

QSGW calculations for point defects using a cut-and-paste approach for the self-energy

The self-energy in quasiparticle self-consistent GW (QSGW) approximation is usually represented in the basis set of density functional theory (DFT) eigenstates. The real space representation of the self-energy matrix obtained by expanding these eigenstates in an atom-centered basis set, such as linearized muffin-tin orbitals, has great advantages.  In this work, we show that the self-energy matrix is relatively short-ranged in the real space representation. This allows us to create a practical “cut-and-paste” method to construct the self-energy matrix of a large system from those of smaller subunits. This method can be used for the study of point-defects in a large supercell by constructing the real space matrix elements Σ_{RL,R’+TL’} in terms of the host and defect Σ, with the latter obtained from a much smaller defect containing cell. In the case of the As_Ga point-defect in GaAs, we show that the defect can already be represented in the 8 atom conventional fcc unit cell. We show that the band structure of a 64 atom defect supercell using the cut-and-paste method is in good agreement with the exact QSGW calculation for the same cell. The defect band position relative to the valence band maximum and dispersion are virtually identical.   

Real-Space Stochastic GW Calculations Benchmark on GW100

Stochastic implementation of GW is a linear scaling method, ideally suited for calculating quasiparticle energies of large systems. This approach uses the stochastic resolution of identity to represent Green's function as a product of a randomly generated orbital at time zero and an evolved random orbital at a later time. It employs real time propagation of stochastic functions to obtain screened coulomb response function. The response function is efficiently stored using stochastic compression. We have implemented the stochastic GW method in real-space density functional theory code PARSEC. We have benchmarked our stochastic GW implementation on GW100 set against the results obtained from the NanoGW code [1]. We find that our results are in good agreement with the results obtained from the NanoGW code.

[1] W. Gao, and J. Chelikowsky, J. Chem. Theory Comput. 15, 5299 (2019)   

Stochastic cRPA calculations of screened interactions in graphene moire states at high pressures

Twisted bilayer graphene (tBLG) hosts correlated electrons in flat bands, stemming from the coupling between mutually rotated monolayers. The coupling is primarily controlled by the twist angle but equally well by the bilayer’s in-plane strain or compression. It was found previously that under angle twist screening plays a crucial role in reducing bare Coulomb interactions as one approaches the magic angle. The pressure behavior of screening is however less investigated.

In this work [1], we apply the stochastic constrained random phase approximation to study the dynamical renormalization of electronic interactions in twisted bilayer graphene (tBLG) characterized by giant supercells cells with ~9000 electrons. We map the correlated subspace on a model Hamiltonian and study the interplay of screening and the on-site interaction at varying pressure. Our results show a striking difference for pressure dependence of screening compared to the previously computed twist angle dependence.   

Efficient Treatment of Dynamical Renormalizations and Multiscale Approaches Through Stochastic Many-body Methods

Efficient numerical implementations enable applying the many-body perturbation theory to realistic systems. Stochastic approaches based on sampling single-particle states, propagators, and interactions are particularly successful in combination with one-shot perturbative corrections. These methods allow simulating nanoscale problems with thousands and tens of thousands of electrons. I will present our work on expanding this framework to (i) enable self-consistent treatments, and (ii) describe more complex quantum phenomena arising from strong interactions of excited states. The new implementation allows to quickly access both diagonal and off-diagonal elements of the self-energy, update the single-particle states, and identify energy regions characterized by strong couplings among quasiparticles. Further, I will present new randomized sampling methods to efficiently compute the dynamical renormalization via stochastic constrained RPA (and beyond RPA) methods within arbitrarily selected strongly interacting subspace. The approach is general and requires only minimal computational resources (in the order of 100s of CPU hours for systems with 10,000 electrons). I will outline the route towards multiscale simulations in which the weakly (and moderately) correlated electronic states are treated by stochastic perturbation theory combined with the embedding of strongly correlated states.    

Making the intractable tractable: GW-BSE calculation of the nature of interlayer and intralayer moiré excitons in large-area moiré superlattices.

Recent experimental measurements have demonstrated signatures of novel exciton states in the moiré superlattices of transition metal dichalcogenide (TMD) bilayer heterostructures. However, the exact nature of the moiré excitons is difficult, and has yet, to be determined experimentally. An accurate description of these excitons requires a completely account of the electron and hole degrees of freedom. First-principles theoretical study of moiré excitons is strongly hindered by the O(N⁴) scaling of the ab initio GW-Bethe Salpeter equation method with the number of atoms (N). To overcome this challenge, we develop a Pristine Unit-cell Matrix Projection (PUMP) method, which speeds up the computation by at least six orders of magnitude. In this method, we leverage the smooth modulation of the electronic wavefunctions in the moiré superlattice to expand them in an efficient (i.e., small) basis of pristine unit-cell wavefunctions. The electron-hole interaction kernel matrix of the Bethe Salpeter equation is similarly expanded in terms of the pristine unit-cell kernel matrix elements. Using this method, we have computed the optical spectra of TMD hetero-bilayers and discovered novel interlayer and intralayer moiré excitons of dramatically different characters.   

Extending the ab initio Bethe-Salpeter equation approach to include phonon screening

Exciton properties are critical to the optoelectronic response of materials, making their theoretical and computational description important to future technological devices. Many-body perturbation theory (MBPT), and specifically the ab initio GW-Bethe-Salpeter (BSE) equation approach [1], is a highly accurate method for predicting exciton properties that has been successfully applied in diverse systems. However, recent work has shown that electronic effects alone are not enough to provide an accurate description of exciton physics, and the effect of phonons can lead to significant renormalization of exciton peak positions [2] and electron-hole interactions [3]. An extension of the GW-BSE method to account for these phonon effects is therefore necessary. Here we present a first-principles approach based on MBPT, building on the framework introduced in prior work [3] to rigorously implement phonon screening effects into the ab initio BSE method. We apply our method to a variety of semiconductors, quantifying the role of phonon screening on exciton binding energies. We also discuss the effect of temperature on phonon screening of excitons, connecting our formulation to model electron-hole interactions that include exciton-phonon coupling, such as the Haken potential.   

Optimally-Tuned Starting Points for One-Shot GW Calculations of Solids

Accurate quasiparticle (QP) band gaps can be obtained via ab initio many-body perturbation theory within the GW approximation, where G is the one-electron Green's function and W is the screened Coulomb interaction. In practice, one-shot G0W0 calculations which do not self-consistently update G or W are often carried out to reduce computational expense and complexity. However, G0W0 calculations exhibit a strong starting-point dependence on the eigensystem used to construct G and W, limiting the predictive power of this approach. Here, we present G0W0 calculations performed using a Wannier-localized, optimally tuned screened range-separated hybrid (WOT-SRSH) starting point. This functional is optimally tuned per system to obey the ionization potential theorem and has been shown to produce band gaps for semiconductors and insulators in excellent agreement with experiment [1]. We show that G0W0@WOT-SRSH leads to QP band gaps and band structures comparable in accuracy to those produced by self-consistent GW schemes for a range of insulators and semiconductors at a reduced cost. We also discuss the sensitivity of G0W0@WOT-SRSH to perturbations in the underlying parameters in the SRSH functional about their optimal values, as well as the implications for optical gaps and spectra.   

Can vertex-corrected perturbative approaches be improved through symmetry breaking?

Many-body perturbation theory (MBPT) approaches are ubiquitous in the study of materials. The most extended such approach is arguably the GW approximation, which includes classical, electrostatic screening on top of a mean-field reference, but neglects multi-quasiparticle interactions. In principle, such terms can be recovered by introducing non-trivial components of the interaction vertex Γ, set to zero in GW. However, it is only recently that systematic (in the Hedin sense) implementations of vertex-corrected GWΓ have started to be applied in ab initio calculations. Hence, important questions regarding the computational and physical nature of these approaches remain open. In this talk, we address one such issue: the impact of the mean-field reference on the MBPT result, particularly regarding symmetry broken (SB) solutions, in the context of self-consistent GWΓ (scGWΓ). The introduction of symmetry breaking formally absent in the true many-body ground state can improve mean-field approximations. However, scGW cannot typically recover the symmetry, resulting in ultimately worse results than the SB mean-field. We test whether it is possible to formulate vertex corrections that recover symmetries at self-consistency, thus allowing the use of SB mean-field reference states.   

Photoemission spectroscopy from the three-body Green's function.

We present an original approach for the calculation of direct and inverse photo-emission spectra from first principles.

The main goal is to go beyond the standard Green's function approaches, such as the GW method, in order to find a good description not only of the quasiparticles but also of the satellite structures, which are of particular importance in strongly correlated materials.

Our method uses as a key quantity the three-body Green's function, or, more precisely, its hole-hole-electron and electron-electron-hole parts.

We show that, contrary to the one-body Green's function, satellites are already present in the corresponding non-interacting Green's function. Therefore, simple approximations to the three-body self-energy, which is defined by the Dyson equation for the three-body Green's function and which contains many-body effects, can still yield accurate spectral functions. In particular, the self-energy can be chosen to be static which could simplify a self-consistent solution of the Dyson equation.

We also show how the one-body Green's function can be retrieved from the three-body Green's function.

We illustrate our approach by applying it to the symmetric Hubbard dimer.   

Validation of Power Series Expansion for Single Particle Green's Function in Model Hamiltonians and Simple Material Systems.

The GW method for single particle Green's function overlooks  self-consistency to achieve computational cost effectiveness. This manifests as incorrect estimation of the effect of collective modes (plasmons) on the quasiparticle and incorrect satelline band energies. The cumulant approximation which was developed to mitigate this is ad hoc, unsystematic and fails at intermediate to strong electron-collective mode coupling. We have developed a general scheme to re-establish self-consistency by post-processing GW calculations by using a power series. We show that we can not only derive the cumulant results but also get exact green’s function for simple multiband model hamiltonians at intermediate couplings. Finally, we use our scheme to produce corrected band-structures in simple crystal systems and compare them to GW and cumulants.   

Electronic properties of crystalline solids from Wannier-localization–based optimal tuning of a screened range-separated hybrid functional

Accurate prediction of fundamental band gaps of crystalline solid-state systems, entirely within density functional theory, has been a long-standing challenge. Previously, we developed a simple and inexpensive method that achieves this by means of nonempirical optimal tuning of the parameters of a screened range-separated hybrid functional [1]. The tuning involves the enforcement of an ansatz that generalizes the ionization potential theorem to the removal of an electron from an occupied state described by a localized Wannier function in a modestly sized supercell calculation. Here we present applications of the method to band gaps of more complex semiconductors and insulators, including halide perovskites and metal oxides, demonstrating quantitative accuracy.

[1] D. Wing et al., PNAS 118, e2104556118 (2021).   

Relativistic Self-ConsistentGW: Exact Two-Component Formalism with One-Electron Approximation

We present a formulation of relativistic self-consistent GW based on the exact two-component formalism with one-electron approximation (X2C1e) and non-relativistic Coulomb interactions. Our theory allows us to study scalar relativistic effects, spin-orbit coupling, and the interplay of relativistic effects with electron correlation. Our all-electron implementation is fully ab initio and requires no pseudopotential constructed from relativistic atomic calculations. We examine the effect of the X2C1e approximation by comparison to DFT calculations and reach excellent agreement with the established four-component theory. The simplicity of X2C1e makes it feasible to construct higher order theories, such as embedding theories, on top of GW.