On equivalence of two formulas of orbital magnetic susceptibility for tight-binding models

TL;DR

This paper proves the equivalence of two formulas for orbital magnetic susceptibility in tight-binding models, clarifying their relationship and the role of surface terms in the calculations.

Contribution

It demonstrates the equivalence of Koshino-Ando's and Gómez-Santos and Stauber's formulas, resolving a discrepancy in their apparent differences.

Findings

The two formulas are mathematically equivalent for tight-binding models.

Surface terms arising from momentum-space integration vanish despite non-periodic primitive functions.

The proof clarifies the theoretical foundation of orbital magnetic susceptibility calculations.

Abstract

We prove that two formulas of the orbital magnetic susceptibility, namely, Koshino-Ando's formula and G\'{o}mez-Santos and Stauber's formula, are equivalent for the tight-binding models. The difference between these two formulas can be written in a form containing the surface terms arising from the momentum-space integration, and we show that the surface terms are exactly vanishing although the primitive functions are seemingly non-periodic with respect to the translation by the reciprocal lattice vector.