Orbital Magnetization of Interacting Electrons

We derive an exact expression for the orbital magnetization of electrons with short-range interactions (such as density-density interactions) in terms of exact zero-frequency response functions of the zero-field system. The result applies to weakly and strongly correlated electrons at zero and finite temperature, provided that the local grand potential density only depends on local thermodynamic parameters. We benchmark the formula for non-interacting and weakly-coupled electrons. To zeroth and first orders in the interaction strength, it agrees with the modern theory of orbital magnetization and its recent generalization to self-consistent Hartree-Fock bands. Our work provides an exact framework of interacting orbital magnetization beyond mean-field treatments, and paves the way for quantitative studies of strongly correlated electrons in external magnetic fields.

💡 Research Summary

The paper presents a rigorous and general formula for the orbital magnetization (OM) of interacting electrons with short‑range density‑density interactions. Starting from a tight‑binding Hamiltonian that includes exponentially decaying hopping amplitudes and interaction matrix elements, the authors construct an auxiliary system in which both the local energy scale and the magnetic field are modulated slowly in space with wave‑vector q ≈ 1/L. By assuming that the grand‑potential density responds locally to these macroscopic modulations, they relate the change in the grand potential of the auxiliary system to the desired uniform‑field OM.

A perturbative expansion of the auxiliary Hamiltonian ΔK̂_B is performed in the infinitesimal parameters η (energy‑scale modulation) and A₀ (vector‑potential amplitude). First‑order contributions vanish because of momentum conservation, leaving only second‑order non‑degenerate perturbations. The key technical step is the use of the Lehmann spectral representation to express the second‑order energy shift as a zero‑frequency principal‑value response function C_P(·Ĥ)(ω = 0). This yields a compact expression for the change in the grand potential, δΩ_B = ½ C_P(ΔK̂_B, ΔK̂_B)(ω = 0).

Taking the long‑wavelength limit (q → 0) and expanding the trigonometric factors, the authors arrive at their central result, Eq. (14). The orbital magnetization is written as a sum over lattice sites and orbital indices of products of hopping matrix elements, coordinate factors (differences of y‑coordinates and sums of x‑coordinates), and two exact zero‑frequency response functions: I, a two‑point electron correlation, and J, a four‑point density‑density correlation. Importantly, all quantities are evaluated in the zero‑field, interacting ground state, so no magnetic field appears explicitly in the formula.

The applicability of Eq. (14) hinges on a locality condition: the response functions I and J must decay faster than |R_C12 − R_C34|⁻². Because the underlying Hamiltonian is short‑ranged, this condition is satisfied for insulators and Fermi liquids in two and three dimensions, both at zero and finite temperature. If the decay is slower, the formula breaks down, signaling non‑local grand‑potential density and the failure of the local‑response picture.

In the non‑interacting limit, I and J reduce to products of free‑electron Green’s functions, and Eq. (14) reproduces the modern theory of orbital magnetization, i.e., the Brillouin‑zone integral involving Berry curvature and orbital moment of Bloch states. For weak interactions, the authors expand I and J to first order in the interaction strength U. The resulting correction separates into three parts: self‑energy, vertex, and pure interaction (J) contributions. They show analytically that the derivative ∂ Ṁ_z/∂U|_{U=0,μ} coincides with the derivative obtained from a self‑consistent Hartree‑Fock calculation, confirming that Eq. (14) correctly captures the first‑order interaction effects already incorporated in static mean‑field theories.

Numerical verification is performed on a QWZ‑like spinless two‑band lattice model. Both a Chern‑insulating phase (C = −1, μ = 0) and a metallic phase (μ = 1.5) are examined on a 40 × 40 lattice. The first‑order interaction derivative computed from the weak‑coupling expansion (k_U) matches the Hartree‑Fock derivative (k_HF) within numerical accuracy, and the antisymmetrized version of Eq. (14) (x↔y with a sign change) yields the same result, confirming the internal consistency of the formalism.

In conclusion, the authors provide an exact, zero‑field‑only expression for orbital magnetization that is valid for a broad class of interacting electron systems, from weakly correlated metals to strongly correlated insulators. The formula bridges the gap between the modern non‑interacting theory and mean‑field extensions, offering a pathway to incorporate genuine many‑body correlations using state‑of‑the‑art numerical techniques (quantum Monte Carlo, tensor‑network methods, cluster DMFT, etc.). This development opens the door to quantitative predictions of orbital magnetization in emerging materials where strong correlations and topology intertwine, such as moiré superlattices, twisted bilayer graphene, and correlated Chern insulators.