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

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.

💡 Research Summary

The paper presents a comprehensive many‑body framework for calculating orbital magnetic responses in interacting multiband electron systems. Starting from the Luttinger–Ward functional, the authors express the thermodynamic potential Ω_int = −T Tr ln G⁻¹ − T Tr(Σ∘G) + Φ(G) and develop a systematic expansion in powers of a weak magnetic field B. The central technical innovation is a modified Fourier transform that separates the gauge‑dependent Wilson line from the translationally invariant part of Green’s functions and self‑energies. This yields gauge‑covariant objects G′ and Σ′ that depend only on relative coordinates and can be represented in an effective momentum space where the magnetic field appears through a non‑commutative Moyal product: (f⋆g)(p)=e^{iθ_{ab}←∂{p_a}→∂{p_b}}f(p)g(p) with θ_{ab}=qB ε_{ab}. The resulting Dyson equation takes the compact form (G₀⁻¹−Σ′)⋆G′=1, making the B‑dependence explicit in the algebraic structure.

To evaluate the thermodynamic potential, the authors reformulate the −T Tr ln G⁻¹ term using a fermionic path integral and the Moyal algebra, thereby avoiding the complications of the Wilson line inside the logarithm. They show that, after expanding the Moyal product, the linear‑in‑B contribution to Ω_int depends only on the zero‑field Green’s function G′(0) and self‑energy Σ′(0). The remaining terms, −T Tr(Σ∘G)+Φ(G), cancel part of the linear contribution, consistent with the stationarity of the Luttinger–Ward functional.

Applying this formalism, the spontaneous orbital magnetization M_orb = −(1/V)∂Ω_int/∂B|{B=0} is derived. The result splits into two gauge‑invariant pieces that are natural generalizations of the orbital magnetic moment and Berry curvature to the combined momentum–frequency space. Defining the renormalized zero‑field Hamiltonian H(0)=h_p+Σ′(0) and its eigenvectors |u_n(p,ω)⟩, the authors introduce a generalized magnetic moment
μ̃_n(p,ω)=½ ε{ab}⟨∂{p_a}u_n|H(0)−ε_n|∂{p_b}u_n⟩
and a generalized Berry curvature
Ω̃_n^{ab}(p,ω)=i⟨∂{p_a}u_n|∂{p_b}u_n⟩−(a↔b).
Both quantities are Hermitian and remain well defined for frequency‑dependent (but Hermitian) self‑energies. The orbital magnetization is expressed as a sum over occupied states of μ̃ and Ω̃ weighted by the Fermi function. When the self‑energy is frequency‑independent (static), μ̃ and Ω̃ reduce to the familiar orbital magnetic moment and Berry curvature of non‑interacting Bloch electrons, and the formula collapses to the standard Středa‑type expression linking magnetization to the Hall conductivity.

The paper also discusses the physical meaning of the non‑commutative coordinates: they correspond to mechanical momenta whose components obey