The $GW$ approximation is the gold standard for calculating quasiparticle band structures, yet its computational cost frequently necessitates the use of the plasmon-pole approximation (PPA). While PPA is known to be highly accurate for simple metals and weakly correlated semiconductors, its regime of validity in materials with competing energy scales remains poorly quantified. Here, we construct an exact, numerical benchmark of the PPA using the Lehmann representation of the density response on finite one-dimensional Hubbard clusters. By analytically convolving the exact non-interacting Green's function $G_0$ with the exact pole representation of the screened interaction $W$, we compute the $GW$ self-energy without numerical frequency integration. We demonstrate that in single-scale Mott insulators, the moment-conserving PPA is essentially exact. However, in multi-band semiconductors where interband transitions introduce a low-energy screening channel that competes with high-energy Mott fluctuations, the PPA systematically misjudges the quasiparticle gap by several electron-volts. Furthermore, we show that strong multi-pole screening can drive the exact quasiparticle weight $Z \to 0$, destroying the quasiparticle picture-an effect entirely missed by the PPA, which artificially stabilizes sharp quasiparticles. Finally, we propose a computationally inexpensive diagnostic based on the polydispersity of the loss function's spectral weight, which accurately predicts PPA failure \textit{ab initio}. Our results provide a rigorous framework for assessing the validity of dynamical screening approximations in strongly correlated materials.
The GW approximation is the gold standard for calculating quasiparticle band structures, yet its computational cost frequently necessitates the use of the plasmon-pole approximation (PPA). While PPA is known to be highly accurate for simple metals and weakly correlated semiconductors, its regime of validity in materials with competing energy scales remains poorly quantified. Here, we construct an exact, numerical benchmark of the PPA using the Lehmann representation of the density response on finite one-dimensional Hubbard clusters. By analytically convolving the exact non-interacting Green's function G0 with the exact pole representation of the screened interaction W, we compute the GW self-energy without numerical frequency integration. We demonstrate that in single-scale Mott insulators, the moment-conserving PPA is essentially exact. However, in multi-band semiconductors where interband transitions introduce a low-energy screening channel that competes with high-energy Mott fluctuations, the PPA systematically misjudges the quasiparticle gap by several electron-volts. Furthermore, we show that strong multi-pole screening can drive the exact quasiparticle weight Z→0, destroying the quasiparticle picture-an effect entirely missed by the PPA, which artificially stabilizes sharp quasiparticles. Finally, we propose a computationally inexpensive diagnostic based on the polydispersity of the loss function's spectral weight, which accurately predicts PPA failure \textit{ab initio}. Our results provide a rigorous framework for assessing the validity of dynamical screening approximations in strongly correlated materials.
Accidental accuracy and vertex corrections in GW: Exact benchmarks for the extended Hubbard model
The GW approximation is the standard tool for quasiparticle predictions in materials, yet its regime of validity in correlated systems remains poorly quantified, because \textit{ab initio} vertex corrections are computationally prohibitive. Using exact diagonalization of the half-filled extended Hubbard model on finite rings as a numerically exact reference, we construct the corresponding model-space GW theory on the identical Hilbert space and quantify its error as a function of local (U) and non-local (V) interaction strength. We find that the required vertex correction changes character across the phase diagram: in the weak-coupling regime the effective vertex Γeff<1, reflecting the suppression of RPA charge fluctuations by exact short-range correlations, whereas in the Mott regime Γeff grows monotonically (to ∼3 at U=8t for N=6), reflecting the local dynamical self-energy structure required to open the Hubbard gap. Vertex corrections in the electron-hole (polarizability) channel are shown to \emph{worsen} the gap error, indicating that reproducing the Mott gap requires dynamical self-energy structure rather than improved screening. For V=0, static COHSEX is accidentally exact at a single crossover U∗≈3.5t; finite V, through non-local Fock exchange, splits this point into a double-crossover window that collapses toward weak coupling. Even at the crossover, however, the exact spectral function retains Hubbard-band structure that no static functional reproduces, so gap agreement does not imply functional accuracy. These results yield quantitative diagnostics for the reliability of GW in correlated materials.
Dynamical screening of the Coulomb interactions in correlated electron systems results in a low-energy effective problem with a dynamical Hubbard interaction $\mathcal{U}(\ensuremath{\omega})$. We propose a Green's function ansatz for the Anderson impurity problem with retarded interactions, in which the Green's function factorizes into a contribution stemming from an effective static-U problem and a bosonic high-energy part introducing collective plasmon excitations. Our approach relies on the scale separation of the low-energy properties, related to the instantaneous static U, from the intermediate- to high-energy features originating from the retarded part of the interaction. We argue that for correlated materials where retarded interactions arise from downfolding higher-energy degrees of freedom, the characteristic frequencies are typically in the antiadiabatic regime. In this case, accurate approximations to the bosonic factor are relatively easy to construct, with the most simple being the boson factor of the dynamical atomic-limit problem. We benchmark the quality of our method against numerically exact continuous-time quantum Monte Carlo results for the Anderson-Holstein model both at half- and quarter-filling. Furthermore, we study the Mott transition within the Hubbard-Holstein model within extended dynamical mean field theory. Finally, we apply our technique to a realistic three-band Hamiltonian for SrVO3. We show that our approach reproduces both the effective mass renormalization and the position of the lower Hubbard band by means of a dynamically screened U, previously determined ab initio within the constrained random phase approximation. Our approach could also be used within schemes beyond dynamical mean field theory, opening a quite general way of describing satellites and plasmon excitations in correlated materials.
In principle, the Luttinger-Ward Green's-function formalism allows one to compute simultaneously the total energy and the quasiparticle band structure of a many-body electronic system from first principles. We present approximate and exact expressions for the correlation energy within the GW-random-phase approximation that are more amenable to computation and allow for developing efficient approximations to the self-energy operator and correlation energy. The exact form is a sum over differences between plasmon and interband energies. The approximate forms are based on summing over screened interband transitions. We also demonstrate that blind extremization of such functionals leads to unphysical results: imposing physical constraints on the allowed solutions (Green's functions) is necessary. Finally, we present some relevant numerical results for atomic systems.
Density functional theory (DFT) performs reasonably well for the determination of structural properties of many materials, but fails to predict electronic band gap values accurately. \nSuch a failure of DFT is not unexpected, since there exists no formal justification for interpreting the DFT eigenvalues as addition or removal energies of the many-body system (quasiparticle energies).\nAn alternative approach to the study of exchange-correlation effects in many-particle systems is provided by many-body perturbation theory (MBPT), which defines a rigorous approach to description of excited-state properties based on the Green's function formalism and the concept of quasiparticle electrons. Within MBPT, one can calculate the quasiparticle (QP) energies and the QP amplitudes by approximately solving, within the so-called GW approximation, the set of coupled integro-differential equations proposed by Hedin in 1965.\nThe GW method generally yields significantly better values for QP energies with respect to DFT as it accounts for the dynamic many-body effects in the electron-electron interaction going beyond the mean-field approximation of independent Kohn-Sham particles. The drawback of such an involved approach, however, is its large computational cost, which is mainly due to the evaluation of dielectric matrices, their inversion, and the solution of a non-Hermitian and nonlinear eigenvalue problem. \nThe first part of this PhD research is devoted to the implementation of an efficient and scalable coarse-grained parallel algorithms and the development of state-of-the-art methods to solve the GW equations.\nAll these techniques are then applied to the study of the quasiparticle band structures of SiO2 in the alpha quartz crystalline structure. The effects of the different approximations involved in the theory, the influence of self-consistency, and a systematic analysis of the reliability of the different plasmon-pole models are presented and discussed. \nThe second part of this PhD work is dedicated to the implementation of a methodological approach to the solution of the GW equations based on the so-called projector augmented wave method (PAW) proposed by Blochl in 1994. This new approach permits to remove many of the limitations intrinsic to the use of the pseudopotential technique widely used for ab-initio calculations, allowing one to reach GW results closer to an all-electron implementation while still maintaining computational efficiency, and flexibility. This allows us to achieve a coherent implementation of many different algorithms which enables a detailed comparison of GW calculations not yet performed until now. Results for the quasiparticle band structure and optical spectra of prototype s- and p-compounds are discussed and compared to recent studies reported in the scientific literature (when available).
Spectral properties from an efficient analytical representation of the GW self-energy within a multipole approximation
We propose an efficient analytical representation of the frequency-dependent GW self-energy Σ via a multipole approximation (MPA-Σ). The multipole-Padé model for the self-energy is interpolated from a small set of numerical evaluations of Σ in the complex frequency plane, similarly to the previously multipole representation developed for the screened Coulomb interaction (MPA-W) [Phys. Rev. B \textbf{104}, 115157 (2021)]. We show that, likewise MPA-W, an appropriate choice of frequency sampling in MPA-Σ is critical to guarantee computational efficiency and high accuracy. The combined MPA-W and MPA-Σ scheme considerably reduces the cost of full-frequency self-energy calculations, especially for spectral band structures over a wide energy range. Crucially, MPA-Σ enables a multipole representation for the interacting Green's function G (MPA-G), providing a straightforward evaluation of all the spectral properties, and a more general way to define the renormalization factor Z. We validate the MPA-Σ and MPA-G approaches for diverse systems: bulk Si, Na and Cu, monolayer MoS2, the NaCl ion-pair and the F2 molecule. Moreover, we introduce toy MPA-Σ/G models to examine the quasiparticle picture in different regimens of weak and strong correlation. With these models, we expose limitations in defining Z from the local derivative of Σ.
Many strongly correlated electron systems fall outside the conventional quasiparticle paradigm. Their low-energy fermionic excitations need not be adiabatically connected to the bare electrons or appear as poles of the physical electron single-particle Green's function. This thesis investigates to what extent an effective quasiparticle description can nevertheless survive, and which emergent degrees of freedom provide the appropriate low-energy structure. First, the interaction-driven evolution from a quantum spin Hall insulator to a Mott insulator is studied using the dynamical cluster approximation. Dispersive zeros of the single-particle Green's function emerge near the Mott transition and retain a nontrivial topological character. When symmetry breaking is allowed, a nontopological excitonic insulator is found to intrude between quantum spin Hall and Mott insulators. Encoding both poles and zeros within a low-energy quasiparticle Hamiltonian provides a continuous one-body description of the three phases. On the Mott side, its gap evolution supports interpreting the soft excitonic mode as a bound state of quasiparticle and quasihole excitations tied to the Green's function zeros, rather than to the widely separated Hubbard bands. Second, the intrinsic anomalous Hall conductivity of a topological metal is analyzed within a multiband extension of Landau Fermi-liquid theory. Residual interactions among quasiparticles at the Fermi surface dress the dynamic current vertex and generate corrections to the Berry-curvature contribution obtained from the quasiparticle bands alone. Our result supports recent claims that the correct expressions for topological observables include vertex corrections besides the topological invariants built just upon the Green's function. It also demonstrates that such corrections are naturally accounted for by Landau Fermi-liquid theory. Extracting the relevant ingredients for an effective description of material-relevant correlated systems requires computational tools that remain reliable at low temperatures. To this end, a deterministic strong-coupling impurity solver for dynamical mean-field theory is discussed. By providing static, dynamical, and thermodynamic observables without stochastic sampling, it offers a promising route toward multiorbital and spin-orbit coupled problems where conventional Monte Carlo methods currently struggle. Finally, the auxiliary quasiparticle framework underlying the ghost-Gutzwiller Ansatz is investigated. On a model for correlated quantum spin Hall insulators, we show that the topology of the interacting state is encoded in the auxiliary one-body Hamiltonian. The resulting band-structure reveals topological Hubbard bands whose topological character can be tuned through a finite magnetization. When applied to the single-band t-J model, the same framework yields a fractionalized Fermi liquid (FL*). In this phase, dispersive neutral spinons, with vanishing physical-electron spectral weight, coexist with conventional quasiparticles forming a small hole-like Fermi surface, thus violating Luttinger's theorem. The corresponding temperature--doping phase diagram contains a low-doping FL*, a d-wave superconducting dome, and an overdoped conventional Fermi liquid, thereby reproducing key qualitative features of cuprate phenomenology.