We propose an efficient analytical representation of the frequency-dependent
GW self-energy $\mathrm{\ensuremath{\Sigma}}$ via a multipole approximation ($\text{MPA-}\mathrm{\ensuremath{\Sigma}}$). The MPA self-energy model is interpolated from a small set of numerical evaluations of $\mathrm{\ensuremath{\Sigma}}$ in the complex frequency plane, similar to the MPA interpolation developed for the screened Coulomb interaction (
MPA-W) [D. A. Leon et al., Phys. Rev. B 104, 115157 (2021)]. Crucially, $\text{MPA-}\mathrm{\ensuremath{\Sigma}}$ enables a multipole representation for the interacting Green's function
G (
MPA-G) and, in turn, access to all the spectral properties, including quasiparticle energies (QP) and renormalization factors beyond the linearized QP equation. We validate the $\text{MPA-}\mathrm{\ensuremath{\Sigma}}$ and
MPA-G approaches for a diverse set of systems: bulk Si, Na, and Cu, monolayer
MoS2, the NaCl ion pair, and the
F2 molecule. We show that, just as for
MPA-W, an appropriate choice of frequency sampling in $\text{MPA-}\mathrm{\ensuremath{\Sigma}}$ is critical to guarantee computational efficiency and high accuracy. Moreover, the combined
MPA-W and $\text{MPA-}\mathrm{\ensuremath{\Sigma}}$ scheme considerably reduces the cost of full-frequency self-energy calculations, especially for spectral band structures over a wide energy range.
The analytic continuation of the GW self-energy from the imaginary to the real-energy axis is a central difficulty for approaches exploiting the favorable properties of response functions at imaginary frequencies. Within a scheme merging contour-deformation and analytic-continuation techniques, we show on the basis of extensive calculations for large molecular sets that it is preferable to perform an analytic continuation of the dynamically screened-Coulomb potential W rather than the much more structured self-energy operator. The case of states lying far away from the gap, including core states, is addressed by generalizing the analytic-continuation scheme, accounting further for quasiparticle lifetimes.
The analytic continuation of the GW self-energy from the imaginary to the real energy axis is a central difficulty for approaches exploiting the favourable properties of response functions at imaginary frequencies. Within a scheme merging contour-deformation and analytic-continuation techniques, we show on the basis of extensive calculations for large molecular sets that it is preferable to perform an analytic continuation of the dynamically screened Coulomb potential
W rather than the much more structured self-energy operator. The case of states lying far away from the gap, including core states, is addressed by generalizing the analytic continuation scheme, accounting further for quasiparticle lifetimes.
We present a new algorithm to analytically continue the self-energy of quantum many-body systems from Matsubara frequencies to the real axis. The method allows straightforward, unambiguous computation of electronic spectra for lattice models of strongly correlated systems from self-energy data that has been collected with state-of-the-art continuous time solvers within dynamical mean-field simulations. Using well-known analytical properties of the self-energy, the analytic continuation is cast into a constrained minimization problem that can be formulated as a quadratic programmable optimization with linear constraints. The algorithm is validated against exactly solvable finite-size problems, showing that all features of the spectral function near the Fermi level are very well reproduced and coarse features are reproduced for all energies. The method is applied to two well-known lattice problems, the two-dimensional Hubbard model at half filling, where the momentum dependence of the gap formation is studied, as well as a multiband model of NiO, for which the spectral function can be directly compared to experiment. Agreement with published results is very good.
The decay properties of the one-particle Green function in real space and imaginary time are systematically studied for solids. I present an analytic solution for the homogeneous electron gas at finite and at zero temperature as well as asymptotic formulas for real metals and insulators that allow an analytic treatment in electronic-structure calculations based on a space-time representation. The generic dependence of the decay constants on known system parameters is used to compare the scaling of reciprocal-space algorithms for the
GW approximation and the space-time method.
In this lecture we present many-body perturbation theory as a method to determine quasiparticle excitations in solids, especially electronic band structures, accurately from first principles.The main ingredient is the electronic self-energy that, in principle, contains all many-body exchange and correlation effects beyond the Hartree potential.As its exact mathematical expression is unknown, approximations must be used in practical calculations.The approximation is obtained using a systematic algebraic approach on the basis of Green function techniques.It constitutes an expansion of the self-energy up to linear order in the screened Coulomb potential, which describes the interaction between the quasiparticles and includes dynamic screening through the creation of exchange-correlation holes around the bare particles.The implementation of the approximation relies on a perturbative treatment starting from density functional theory.Besides a detailed mathematical discussion we focus on the underlying physical concepts and show some illustrative applications.