# Energies and spectra of solids from the algorithmic inversion of dynamical Hubbard functionals

## TL;DR

This paper introduces an efficient algorithmic approach to compute spectral and thermodynamic properties of solids using dynamical Hubbard functionals, simplifying complex frequency-dependent calculations and achieving results comparable to advanced methods.

## Contribution

The authors develop a novel pole-sum representation and map the Dyson equation into an exactly solvable nonlinear eigenvalue problem, enhancing computational efficiency and physical insight.

## Key findings

- Accurate spectral and thermodynamic properties of SrVO3 match experimental data.
- Significant reduction in computational cost compared to traditional methods.
- Method provides a transparent physical interpretation of dynamical effects.

## Abstract

Energy functionals of the Green's function can simultaneously provide spectral and thermodynamic properties of interacting electrons' systems. Though powerful in principle, these formulations need to deal with dynamical (frequency-dependent) quantities, increasing the algorithmic and numerical complexity and limiting applications. We first show that, when representing all frequency-dependent propagators as sums over poles, the typical operations of dynamical formulations become closed (i.e., all quantities are expressed as sums over poles) and analytical. Importantly, we map the Dyson equation into a nonlinear eigenvalue problem that can be solved exactly; this is achieved by introducing a fictitious non-interacting system with additional degrees of freedom which shares, upon projection, the same Green's function of the real system. Last, we introduce an approximation to the exchange-correlation part of the Klein functional adopting a localized $GW$ approach; this is a generalization of the static Hubbard extension of density-functional theory with a dynamical screened potential $U(\omega)$. We showcase the algorithmic efficiency of the methods, and the physical accuracy of the functional, by computing the spectral, thermodynamic, and vibrational properties of SrVO$_3$, finding results in close agreement with experiments and state-of-the-art methods, at highly reduced computational costs and with a transparent physical interpretation.