An Energy Integration Free Kubo-Bastin Formula Decomposition.

Kubo formulae play a central role in modern spintronics and condensed matter physics, serving
as the foundational ground for studying transport responses in the linear regime. In this work, we
propose a reformulation of the widely used Kubo-Bastin decompositions that eliminates the need
for numerical energy integration. By performing these integrations analytically for generic periodic
systems, our approach significantly reduces computational cost and simplifies the calculation of
transport coefficients, while maintaining a clear distinction between the relevant Fermi-surface and
Fermi-sea transport components.