A Quantum Many-Body Approach for Orbital Magnetism in Correlated Multiband Electron Systems

TL;DR

This paper develops a quantum many-body framework to accurately compute orbital magnetic responses in correlated multiband electron systems, extending geometric formulas to include interactions.

Contribution

It introduces a novel formalism based on the Luttinger-Ward functional and noncommutative coordinates, enabling analytic calculations of orbital magnetism in interacting systems.

Findings

Derived the spontaneous orbital magnetization in correlated systems.

Generalized orbital magnetic moment and Berry curvature to momentum-frequency space.

Reduced to known formulas for noninteracting systems when self-energy is frequency-independent.

Abstract

Orbital magnetism is a purely quantum phenomenon that reflects intrinsic electronic properties of solids, yet its microscopic description in interacting multiband systems remains incomplete. We develop a general quantum many-body framework for orbital magnetic responses based on the Luttinger-Ward functional. Starting from the Dyson equation, we reformulate the thermodynamic potential in a weak magnetic field and construct a controlled expansion in powers of $B$ applicable to correlated electron systems. A key technical advance is a modified ``Fourier'' representation using noncommutative coordinates, which allows the thermodynamic potential to be expressed in an effective momentum space where the magnetic field acts perturbatively. This formulation makes analytic progress possible within the Moyal algebra. As an application, we derive the spontaneous orbital magnetization and express it entirely in terms of the zero-field Hamiltonian renormalized by the self-energy. For frequency-dependent but Hermitian self-energies, we generalize the orbital magnetic moment and Berry curvature to momentum-frequency space and identify two gauge-invariant contributions built from these quantities. For frequency-independent self-energies the result reduces to the familiar geometric formula for noninteracting systems. This framework provides a unified foundation for computing orbital magnetic responses in correlated multiband materials.