Theory of local orbital magnetization: local Berry curvature

Technical Summary: Theory of Local Orbital Magnetization and Local Berry Curvature

Problem Statement
While a consistent quantum theory of orbital magnetization for periodic crystalline solids was established in 2005 (defining magnetization via orbital moments and Berry curvature), a microscopic theory for local orbital magnetization valid across diverse geometries has remained absent. Existing frameworks, such as the modern theory for crystals, rely on unit-cell averages, obscuring sub-lattice and single-site magnetic textures. Furthermore, while the Středa formula links orbital magnetization to Hall conductivity in periodic systems, it is unclear how topological information is encoded locally in finite systems, ribbons, or non-crystalline structures. Previous real-space approaches, such as the Bianco-Resta local Chern marker, define local magnetization starting from the global total magnetization, introducing gauge ambiguities related to surface currents and hinge effects. This work addresses the need for a unified, gauge-invariant thermodynamic theory that resolves orbital magnetization at the atomic scale across periodic crystals, ribbons, and finite open-boundary systems.

Methodology
The authors develop a thermodynamic theory based on a perturbative expansion of the magnetic-field-dependent local density of states (LDOS), $\rho(r, \varepsilon, B)$.

• Formalism: Starting from the local grand potential $\Xi(r, \mu, B) = -\int d\varepsilon F(\varepsilon)\rho(r, \varepsilon, B)$, the local orbital magnetization is defined as the zero-field limit $M(r, \mu) = -\partial_B \Xi(r)|_{B=0}$.
• Model: The theory is formulated for non-interacting spinless electrons in a tight-binding model, incorporating the magnetic field via the Peierls substitution. The LDOS is evaluated using gauge-invariant perturbation theory.
• Derivation: The central result is an expression for local magnetization involving the retarded Green's function $G(\varepsilon)$ and the velocity operator $\hat{v}$:
  $$M_{orb}(r, \mu) = \frac{e}{2\pi\hbar} \int d\varepsilon F(\varepsilon) \text{Re}\langle r | G [(\hat{v}G) \times (\hat{v}G)] | r \rangle$$
  This expression is evaluated for three distinct geometries: finite systems with open boundaries, infinite periodic crystals (sublattice resolved), and ribbon geometries.

Key Contributions and Results

1. Identification of Local Berry Curvature:
   The theory reveals a previously unidentified quantity: a local Berry curvature, $\Omega_n(r)$ (for finite systems) or $\Omega_{nk}(r_\alpha)$ (for sublattices).

   • In finite systems, the local magnetization separates into two terms: a projected orbital moment $m_n(r) = |\langle r|n\rangle|^2 m_n$ and the local Berry curvature term $F(\varepsilon_n)\Omega_n(r)$.
   • Physically, while the orbital moment describes the magnetic-field-induced shift of energy levels (independent of position), the local Berry curvature governs the field-induced redistribution of electronic spectral weight in real space. It satisfies a sum rule $\sum_r \Omega_n(r) = 0$, indicating it redistributes weight without changing the total normalization of the state.

2. Unified Description Across Geometries:
   The formalism provides a consistent description for:

   • Finite Systems: The local magnetization recovers the conventional total orbital magnetization upon summation over all sites, as the local Berry curvature contributions cancel out.
   • Periodic Crystals: The theory resolves magnetization on a sublattice scale. The local Berry curvature decomposes into the projection of the conventional Bloch Berry curvature and a purely geometric contribution, $\Omega_{nk}^{geom}(r_\alpha)$. This geometric term redistributes spectral weight among sublattices within a unit cell and satisfies $\sum_\alpha \Omega_{nk}^{geom}(r_\alpha) = 0$.
   • Ribbons: The same separation into orbital moments and local Berry curvature holds, providing a unified framework for all three geometries.

3. Topological Encoding and Slope Quantization:
   The authors demonstrate that the slope of the local orbital magnetization within an insulating gap is quantized and satisfies the Středa relation $\partial_\mu M = \frac{e}{h}C$ even locally in ribbon and finite geometries.

   • This slope is governed by the local Berry curvature.
   • In two-band systems, the slope comprises a sublattice-independent topological contribution (half the Chern number) and a sublattice-dependent geometric contribution arising from $\Omega_{nk}^{geom}$.
   • The geometric contribution vanishes in particle-hole symmetric systems but is significant in trivial gaps, distinguishing different orbital magnetic orders (ferro-, antiferro-, and ferrimagnetic) at the sublattice level.

4. Comparison with Bianco-Resta Approach:
   The paper contrasts its results with the Bianco-Resta local Chern marker.

   • Computational Efficiency: The new formulation scales as $N^2$ compared to $N^3$ for Bianco-Resta.
   • Conceptual Difference: The Bianco-Resta approach constructs local magnetization from the global total, leading to gauge ambiguities associated with surface magnetization. The current theory derives the local response directly from the field-dependent LDOS, providing a unique, gauge-invariant definition of local magnetization.
   • Physical Distinction: The two approaches yield markedly different sublattice textures. Crucially, the Bianco-Resta approach fails to account for the slope of magnetization in the gap related to the geometric part of the local Berry curvature, which is prominent in trivial phases.

5. Anatomy of Bulk Topology:
   The theory decomposes the topological response into contributions from band states and gap states.

   • Locally (in the bulk): The topological response originates entirely from the local Berry curvature of band states; gap states do not contribute to the local bulk topological response.
   • Macroscopically: The quantized slope is recovered through different microscopic mechanisms depending on geometry. In ribbons, it is carried by band-state Berry curvature. In finite open-boundary systems, the Berry curvature contribution vanishes upon summation, and the quantized slope is recovered through the orbital moments of gap states.

Significance
The paper claims to establish local orbital magnetization as a genuine microscopic observable, extending the modern theory of orbital magnetization from a unit-cell averaged quantity to the atomic scale. By identifying the local Berry curvature, the work provides a bulk description of topology in finite systems and reveals that topological responses can be encoded locally in the bulk independent of edge states. The framework resolves orbital magnetic textures at the single-site and sublattice scales, offering a unified thermodynamic perspective on how distinct microscopic mechanisms (orbital moments vs. Berry curvature redistribution) cooperate to produce macroscopic quantized responses. This opens the door to studying orbital magnetic textures in systems lacking translational symmetry, such as molecules, amorphous materials, quasicrystals, and moiré structures.