Electrons moving in a magnetic field experience the Lorentz force, causing them to move in circular paths. This creates a magnetic moment that opposes the field, representing the classical orbital magnetic susceptibility for free electrons.

In solids, however, electrons are subject to the Coulomb force from periodically aligned atoms. As a result, they adopt Bloch wave functions instead of simple circular orbits. Additionally, impurities can scatter electrons, and some electrons remain tightly bound to atoms. These complexities have made orbital magnetic susceptibility in solids a fundamental yet difficult problem.

While many attempts have been made to develop compact expressions for orbital magnetic susceptibility, most of them are too complex for practical use. The simple Fukuyama formula, which uses Green’s functions, is a key exception. However, its Bloch representation is complicated to solve, and some studies argue that it is inapplicable to multi-band tight-binding models, such as graphene.

In a recent study published in the Journal of the Physical Society of Japan, researchers presented an exact formula of orbital magnetic susceptibility in terms of Bloch wave functions. Starting from the Fukuyama formula, they derived a simple formula containing only four contributions: the Landau–Peierls susceptibility, interband contributions, Fermi-surface contributions, and contributions from occupied states. Due to its fundamental contributions, this research has been honored with ‘The Outstanding Paper Award of the Physical Society of Japan’.

They also clarified the physical meaning of each contribution and demonstrated consistency with results from previous studies. Moreover, by applying this formula to the problem of linear combination of atomic orbitals, they elucidated the itinerant nature of Bloch electrons for the first time.

This unified formula can be directly applied to any kind of solid, including multi-band tight banding models like graphene and bismuth.

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