Symmetric Helmholtz Fermi-surface harmonics for an optimal representation of anisotropic quantities on the Fermi surface: Application to the electron-phonon problem

TL;DR

This paper introduces a symmetry-aware numerical method to efficiently represent anisotropic quantities on the Fermi surface, significantly reducing computational effort in electron-phonon and superconductivity calculations.

Contribution

The authors develop a symmetry-adapted Helmholtz Fermi-surface harmonics basis, enabling optimal and compact representation of anisotropic properties on the Fermi surface.

Findings

Successfully identified symmetric harmonics for FCC-Cu, HEX-MgB2, and BCC-YH6.

Achieved high-accuracy representation of the electron-phonon mass-enhancement parameter λ_k.

Demonstrated potential to drastically reduce computational costs in Fermi surface dependent calculations.

Abstract

We outline a numerical procedure to incorporate the crystal symmetries in the Helmholtz Fermi-surface harmonics basis set, which are the solutions of the Helmholtz equation defined on the Fermi surface. This improvement allows for an optimal representation of anisotropic quantities defined on the Fermi surface in terms of few symmetric elements of the set. We demonstrate the general validity of our approach by identifying the fully symmetric Helmholtz Fermi-surface harmonics subset for several representative systems with different crystal structures, namely, FCC-Cu, HEX-MgB$_2$, and BCC-YH$_6$. Furthermore, we illustrate the potential of the method applied to the electron-phonon problem, showing that the anisotropic electron-phonon mass-enhancement parameter $\lambda_{\bf k}$ can be represented to high accuracy by a handful of coefficients. This works as an effective filter, paving the way for a reduction of several orders of magnitude in the computation of superconductivity, impurity problems, or any other Fermi surface dependent property of metals from first principles.