# Efficient evaluation of the polarization function in dynamical mean-field theory

## TL;DR

This paper presents a fast and accurate method for evaluating the polarization function in dynamical mean-field theory, reducing computational costs and improving the description of critical behavior in correlated electron systems.

## Contribution

The authors introduce an exact decomposition of the DMFT polarization function, enabling efficient calculation of the lattice response with minimized cutoff errors and improved critical behavior analysis.

## Key findings

- The new method reduces frequency summation complexity.
- It accurately captures critical behavior near antiferromagnetic transitions.
- Application to Hubbard models demonstrates improved results.

## Abstract

The dynamical susceptibility of strongly correlated electronic systems can be calculated within the framework of the dynamical mean-field theory (DMFT). The required measurement of the four-point vertex of the auxiliary impurity model is however costly and restricted to a finite grid of Matsubara frequencies, leading to a cutoff error. It is shown that the propagation of this error to the lattice response function can be minimized by virtue of an exact decomposition of the DMFT polarization function into local and nonlocal parts. The former is measured directly by the impurity solver, while the latter is given in terms of a ladder equation for the Hedin vertex that features an unprecedentedly fast decay of frequency summations compared to previous calculation schemes, such as the one of the dual boson approach. At strong coupling the local approximation of the TRILEX approach is viable, but vertex corrections to the polarization should be dropped on equal footing to recover the correct prefactor of the effective exchange. In finite dimensions the DMFT susceptibility exhibits spurious mean-field criticality, therefore, a two-particle self-consistent and frequency-dependent correction term is introduced, similar to the Moriya-$\lambda$ correction of the dynamical vertex approximation. Applications to the two- and three-dimensional Hubbard models on the square and cubic lattices show that the expected critical behavior near an antiferromagnetic instability is recovered.