In the absence of spin-orbit coupling the Zeeman effect is isotropic in the direction of the magnetic field and leads to Paul paramagnetism.

The orbital motion, leads to Landau diamagnetism, and free electrons with a parabolic dispersion (and mass m) in three dimensions the magnitude is one-third (and opposite in sign) to that of Pauli susceptibility.

What happens for a parabolic band with effective mass m*?

The Pauli susceptibility is enhanced by m*/m and the Landau susceptibility is reduced by m/m*. Thus in semiconductors (where m* can be much less than m) the latter can become dominant.

In a simple Fermi liquid enhancing the interactions will make the spin susceptibility even more dominant over the orbital susceptibility.

What happens in the presence of a band structure?

This problem was "solved" by Peierls in 1933, leading to this formula.

Aside: this is the paper where he introduced the famous Peierls factor for effect of an orbital magnetic field on a tight-binding Hamiltonian.

However, according to two recent papers there is more to the story.

Geometric orbital susceptibility: quantum metric without Berry curvature

Frédéric Piéchon, Arnaud Raoux, Jean-Noël Fuchs, Gilles Montambaux

Orbital Magnetism of Bloch Electrons I. General Formula

Masao Ogata and Hidetoshi Fukuyama

**What happens in the presence of electron-electron interactions?**

This is the question I am ultimately interested in. Clearly in a Fermi liquid regime, strong correlations will enhance m*/m (or equivalently reduce the effective band width) and reduce the relative importance of the orbital susceptibility. However, in other regimes such as in bad metal it is not clear. One treatment of the effect of spin fluctuations is here.

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