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arXiv:2601.14372

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

Mengxing Ye

Published Jan 20, 2026• 0 citations
arXivHTMLPDF

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.

DOI
10.48550/arxiv.2601.14372

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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 BBB 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.
We demonstrate that nontrivial multiband topological invariants of electronic wavefunctions can be revealed through orbital magnetization responses to external magnetic fields. We find that decomposing orbital magnetization into energetic and quantum-geometric contributions allows one to deduce nontrivial multiband topology, provided knowledge of the energy spectrum. We showcase our findings in general effective models with multiband Euler topology. We moreover identify such multiband topological invariants in effective models of strontium ruthenate (Sr2RuO4\text{Sr}_2 \text{Ru} \text{O}_4Sr2​RuO4​), which may in principle be verified in the state-of-the-art doping-dependent magnetization measurements. Our reconstruction scheme for multiband invariants sheds a topological perspective on the multiorbital effects in materials realizing unconventional phenomenologies of orbital currents or multiband superconductivity.
Starting from the full many-body Hamiltonian of interacting electrons the effective self-energy acting on electrons residing in a subspace of the full Hilbert space is derived. This subspace may correspond to, for example, partially filled narrow bands, which often characterize strongly correlated materials. The formalism delivers naturally the frequency-dependent effective interaction (the Hubbard U) and provides a general framework for constructing theoretical models based on the Green's function language. It also furnishes a general scheme for first-principles calculations of complex systems in which the main correlation effects are concentrated on a small subspace of the full Hilbert space.
Abstract Green-function techniques are among the most versatile tools for investigating quantum many-body systems at finite temperature. Yet, standard textbook treatments often fragment the formalism—separating bosons and fermions, systems with massless and massive particles, or restricting proofs to special Hamiltonians—thereby obscuring the structural unity of the theory. This work develops a general, Hamiltonian-independent framework for both real- and imaginary-time Green-functions, valid for any combination of bosonic and fermionic degrees of freedom, including mixed systems such as interacting electron–phonon models. The key conceptual advance is the nature index η , which complements the conventional statistical sign ζ . While ζ encodes particle statistics, η distinguishes canonical quantization of systems with massless particles ( η = 0) from second quantization of systems with massive particles ( η = 1). This classification removes ambiguities in chemical-potential terms and unifies the treatment of photons, phonons, electrons, and their mixtures under a single notation. From this basis, we formulate generalized ensemble constructions, generalized Lehmann-representations, and a fluctuation-dissipation theorem valid for arbitrary Hamiltonians. Several results—such as the monotonicity of particle number in the chemical potential, an explicit formula for the quadratic contribution to the internal energy from the retarded Green-function, and the Hartree-contribution to the self-energy in electron–phonon systems—are either absent from, or only sparsely mentioned in, existing literature. We further introduce Green’s Chart, a visual and algebraic map that systematically connects all Green-function types through analytic continuation, Fourier-transforms, and Wick-rotations, enabling all to be generated from the Matsubara Green-function. Applications to both non-interacting and interacting models demonstrate the framework’s ability to compute densities of states, thermodynamic potentials, and spectral functions with a consistent formalism. The results provide a complete and reproducible toolkit for equilibrium many-body theory, consolidating scattered results into a unified structure suitable for both pedagogical and computational purposes.
We present a construction of a mean-field theory for thermodynamic and spectral properties of correlated electrons reliable in the strong-coupling limit. We introduce an effective interaction determined self-consistently from the reduced parquet equations. It is a static local approximation of the two-particle irreducible vertex, the kernel of a potentially singular Bethe-Salpeter equation. The effective interaction enters the Ward identity from which a thermodynamic self-energy, renormalizing the one-electron propagators, is determined. The dynamical Schwinger-Dyson equation with the thermodynamic propagators is then used to calculate the spectral properties. The thermodynamic and spectral properties of correlated electrons are in this way determined on the same footing and in a consistent manner. Such a mean-field approximation is analytically controllable and free of unphysical behavior and spurious phase transitions. We apply the construction to the asymmetric Anderson impurity and the Hubbard models in the strong-coupling regime.
We construct a semiclassical theory for electrons in a non-Hermitian periodic system subject to perturbations varying slowly in space and time. We derive the energy of the wavepacket to first order in the gradients of the perturbations. Applying the theory to the specific case of a uniform external magnetic field, we obtain an expression for the orbital magnetization energy. Using the principles of non-Hermitian dynamics, we define a physically meaningful non-Hermitian generalization of the angular momentum operator and show that it is compatible with the real part of the orbital magnetic moment. The imaginary part of the orbital magnetic moment is also discussed and shown to originate from an imaginary counterpart to the angular momentum that gives rise to a non-Hermitian generalization of the Aharonov-Bohm effect.
Motivated by recent work on Hall viscosity, we derive from first principles the Kubo formulas for the stress-stress response function at zero wave vector that can be used to define the full complex frequency-dependent viscosity tensor, both with and without a uniform magnetic field. The formulas in the existing literature are frequently incomplete, incorrect, or lack a derivation; in particular, Hall viscosity is overlooked. Our approach begins from the response to a uniform external strain field, which is an active time-dependent coordinate transformation in ddd space dimensions. These transformations form the group GL(d,R)(d,\mathbb{R})(d,R) of invertible matrices, and the infinitesimal generators are called strain generators. These enable us to express the Kubo formula in different ways, related by Ward identities; some of these make contact with the adiabatic transport approach. The importance of retaining contact terms, analogous to the diamagnetic term in the familiar Kubo formula for conductivity, is emphasized. For Galilean-invariant systems, we derive a relation between the stress response tensor and the conductivity tensor that is valid at all frequencies and in both the presence and absence of a magnetic field. In the presence of a magnetic field and at low frequency, this yields a relation between the Hall viscosity, the q2{q}^{2}q2 part of the Hall conductivity, the inverse compressibility (suitably defined), and the diverging part of the shear viscosity (if any); this relation generalizes a result found recently by others. We show that the correct value of the Hall viscosity at zero frequency can be obtained (at least in the absence of low-frequency bulk and shear viscosity) by assuming that there is an orbital spin per particle that couples to a perturbing electromagnetic field as a magnetization per particle. We study several examples as checks on our formulation. We also present formulas for the stress response that directly generalize the Berry (adiabatic) curvature expressions for zero-frequency Hall conductivity or viscosity to the full tensors at all frequencies.