🔬 materials science

Orbital Hall effect from the Wannier function approach

This paper presents a gauge-covariant formalism based on Wannier functions to rigorously resolve the position-operator ambiguity in Bloch states, thereby establishing a transparent theory for calculating orbital angular momentum and orbital Hall conductivity, which is validated through exact finite-lattice calculations of the Haldane model.

Original authors: Tapesh Gautam, S. Satpathy

Published 2026-11-04

📖 4 min read☕ Coffee break read

CC BY 4.0

Original authors: Tapesh Gautam, S. Satpathy

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the microscopic world of solid materials, electrons do not merely flow like water through a pipe; they carry a subtle, intrinsic spin and a swirling motion known as orbital angular momentum. This swirling motion is a fundamental property of how electrons orbit the atomic nuclei within a crystal, much like a planet rotating around a star. For decades, physicists have been fascinated by how this orbital motion can be manipulated to generate electric currents, a field of study that promises new ways to store and process information. One of the most intriguing phenomena in this realm is the orbital Hall effect, where an electric current flowing through a material generates a sideways flow of this orbital motion, creating a current of angular momentum without the need for a magnetic field. Understanding this effect is crucial because it could lead to faster, more efficient electronic devices that rely on the orbital properties of electrons rather than just their charge or spin. However, describing this motion mathematically has been a persistent puzzle, as the standard tools used to model electrons in crystals struggle to define exactly where an electron is located within the repeating pattern of a solid.

A team of researchers at the University of Missouri has now resolved this long-standing ambiguity by developing a new, rigorous way to calculate the orbital angular momentum of electrons in solids. They approached the problem by shifting their perspective from the abstract, repeating patterns of the crystal to a more concrete description using localized "Wannier functions." Think of these functions as a way to pin down the electron's position to a specific spot in the crystal lattice, rather than letting it blur across the entire structure. By using this method, the team created a clear and consistent mathematical framework that defines the orbital angular momentum operator without the confusion that plagued previous attempts. Their work provides a transparent theory for orbital transport, allowing scientists to calculate how these orbital currents behave with a level of precision that was previously impossible.

To prove that their new theory worked, the researchers tested it against a well-known model of a two-dimensional crystal called the Haldane model. In this test, they performed two separate calculations and compared the results. The first calculation used their new, complex formula derived from the Wannier function approach. The second calculation was a direct, brute-force simulation of a finite crystal lattice, consisting of a grid of 60 by 60 atoms, where they calculated the orbital angular momentum directly from its basic physical definition in real space. The results from these two completely different methods matched perfectly. This agreement confirmed that their new formula correctly captures the physics of the system, including the subtle off-diagonal elements that describe how the orbital motion of electrons in one energy band interacts with electrons in another.

The study also clarified why previous methods had failed or produced conflicting results. Earlier approaches often relied on an approximation that treated the orbital motion as if it were centered strictly on the atoms themselves. While this worked for some materials, it completely broke down for others, such as graphene, where the atoms have no intrinsic orbital motion, yet the material still exhibits a strong orbital Hall effect. The new theory correctly predicts that even in these cases, the orbital angular momentum is non-zero and can be calculated accurately. Furthermore, the researchers showed that their results hold true for both insulating materials and metals, demonstrating the broad applicability of their approach.

The implications of this work extend beyond just calculating the orbital Hall effect. The researchers noted that their formalism is equally relevant for other emerging phenomena in the field of orbitronics, such as the orbital Edelstein effect, where an electric field creates a non-equilibrium accumulation of orbital moments, and the generation of orbital torque, which can be used to switch magnetic states in memory devices. By providing a reliable and unambiguous way to define and calculate orbital angular momentum, this work lays a solid foundation for future discoveries in how we can harness the orbital properties of electrons for next-generation technologies. The researchers validated their findings through exact calculations on a specific model, ensuring that the theory is not just a mathematical exercise but a tool that accurately reflects the behavior of electrons in real solids.

Technical Summary: Orbital Hall Effect from the Wannier Function Approach

Problem Statement
The theoretical description of orbital transport phenomena, specifically the orbital Hall effect (OHE) and orbital angular momentum (OAM) in solids, has been hindered by the ill-defined nature of the position operator ( $\mathbf{r}$ ) within the Bloch state representation. While early theoretical works often relied on the "atom-centered approximation" (ACA), replacing the OAM with atom-based angular momentum operators, this approach is fundamentally unregulated. It fails for materials with zero atomic angular momentum, such as graphene, and yields conflicting results when applied to momentum-space approaches. The core difficulty lies in calculating the matrix elements $\langle \psi_{n\mathbf{k}} | \mathbf{r} \times \mathbf{v} | \psi_{n'\mathbf{k}} \rangle$ , where the position operator's ambiguity leads to gauge-dependent results for observable quantities.

Methodology
To resolve the position-operator ambiguity, the authors develop a gauge-covariant formalism for the OAM operator and orbital Hall conductivity (OHC) using the Wannier function approach. Unlike the Bloch representation, the position operator is well-defined in the Wannier representation.

The derivation proceeds as follows:

Symmetric Operator Definition: The OAM operator is defined in a symmetric form, $\mathbf{L} = \frac{m_e}{2}(\mathbf{r} \times \mathbf{v} - \mathbf{v} \times \mathbf{r})$ , to ensure hermiticity.
Basis Transformation: The formalism transforms Bloch states $|u_{n\mathbf{k}}\rangle$ into the Wannier function basis $|n\mathbf{R}\rangle$ . This allows the position operator $\mathbf{r}$ to be expressed in terms of lattice vectors $\mathbf{R}$ and derivatives of the cell-periodic Bloch functions with respect to crystal momentum ( $\partial_\mathbf{k}$ ).
Gauge-Covariant Derivation: By evaluating the matrix elements in the Wannier basis and utilizing translational symmetry, the authors derive an exact expression for the OAM matrix elements $L_{nn'}(\mathbf{k})$ . This expression explicitly incorporates the Berry connection $\mathbf{A}_{n\mathbf{k}} = i\langle u_{n\mathbf{k}} | \partial_\mathbf{k} u_{n\mathbf{k}} \rangle$ to ensure gauge covariance.
Validation via Finite-Lattice Calculation: To validate the derived analytical expression, the authors compute the OAM and OHC for the Haldane model on a honeycomb lattice using two independent methods:
- Method A: Using the derived bulk expression (Eq. 16).
- Method B: Direct numerical evaluation of the OAM operator from its real-space definition on a finite $60 \times 60$ lattice with periodic boundary conditions (PBC). In this method, off-diagonal position matrix elements are derived exactly from the velocity matrix elements, avoiding the ill-defined diagonal terms by utilizing the symmetric form of the operator.

Key Contributions and Results

Derivation of Gauge-Covariant OAM: The paper presents a rigorous, gauge-invariant expression for the OAM operator (Eq. 16) that resolves the ambiguities inherent in Bloch-based treatments. The final expression includes both band-diagonal and off-diagonal terms, the latter being crucial for transport calculations via the Kubo formula.
Resolution of ACA Failure: The authors demonstrate that the derived formalism yields non-zero OAM matrix elements for the Haldane model (an $s$ -orbital system), whereas the atom-centered approximation predicts a vanishing orbital moment.
Correction of Previous Momentum-Space Results: The derived off-diagonal terms differ from some previous momentum-space formulations (e.g., Ref. [26]), which predicted zero off-diagonal OAM for two-band systems. The authors' results, which agree with Ref. [27], show that non-zero off-diagonal elements exist and are essential.
Numerical Validation: The study establishes the validity of the new formalism by showing complete agreement between the bulk Wannier-based expression and the exact real-space finite-lattice calculations for the Haldane model. This agreement holds for both OAM matrix elements and the resulting Orbital Hall Conductivity (OHC) across different topological phases (trivial and non-trivial Chern numbers).
Symmetry Analysis: The derived expression correctly satisfies the required symmetry conditions under time-reversal ( $T$ ) and inversion ( $I$ ) symmetries.

Significance and Claims
The paper claims to provide a "transparent and rigorous theory of orbital transport" by eliminating the position-operator ambiguity through the Wannier function formulation. The significance of this work lies in its ability to:

Provide a consistent framework for calculating orbital transport phenomena (OHE, orbital Edelstein effect, orbital torque) in both metals and insulators.
Correctly describe orbital responses in materials where atomic angular momentum is zero or where previous approximations fail.
Validate the theoretical framework through exact numerical benchmarks, ensuring that the derived expressions are not merely formal but physically accurate.

The authors conclude that their formalism is essential for accurately describing orbital phenomena in solids, particularly where the OAM plays a crucial role, and that the agreement between the two independent calculation methods for the Haldane model firmly establishes the correctness of their derived OAM expression.

More like this