Orbital-driven emergent transport in altermagnets

Abstract: Altermagnets have recently emerged as a promising platform for spintronics due to their unique magnetic symmetry. However, most studies have focused on spin degrees of freedom, leaving the dynamic role of orbital degrees of freedom largely unexplored. In this work, we extend the altermagnet Hamiltonian to include the orbital degree of freedom as a dynamical variable and derive the resulting emergent electromagnetic fields (EEMFs). This approach allows us to demonstrate emergent electric fields controllable via lattice anisotropy and the resulting orbital and magnetic multipole currents. Furthermore, we show that non-vanishing emergent electric fields can arise even in simplified spin and orbital textures, particularly in the presence of dynamic lattice distortion. This formalism is generalizable to high-order altermagnets beyond d-wave systems.

• The paper demonstrates that incorporating dynamic orbital degrees of freedom yields emergent electromagnetic fields that drive novel orbital, charge, and multipole currents.
• It establishes that lattice anisotropy and distortions are critical for generating distinct transport responses, as quantified by anisotropy corrections and Hall conductivities.
• The work proposes experimental approaches using strain and optical methods to validate orbitronics, paving the way for advanced multipolar electronic applications.

Orbital Emergent Transport in Altermagnets: A Comprehensive Analysis

Introduction

This work extends the theoretical understanding of altermagnets by incorporating explicit orbital degrees of freedom into the emergent transport formalism. Previous studies have primarily addressed spin-related phenomena, overlooking the dynamic orbital contributions despite the symmetry constraints and band structures that allow for rich orbital physics. The authors develop a minimal yet generalizable approach that reveals how dynamic orbital textures and lattice distortions yield emergent electromagnetic fields (EEMFs) capable of driving novel charge, spin, orbital, and magnetic multipole currents. The framework not only provides criteria for the occurrence of such phenomena but elucidates their dependence on band anisotropy and symmetry in a manner not accessible in conventional magnetically ordered systems.

Hamiltonian Construction and Emergent Electromagnetic Fields

A central advance of this work is the extension of the altermagnet Hamiltonian to include both spin and orbital N\'eel vectors, denoted $n(\mathbf{r}, t)$ and $l(\mathbf{r}, t)$, respectively. The explicit orbital basis, e.g., $\{\ket{p_x}, \ket{p_y}\}$ or $\{\ket{d_{3z^2-r^2}}, \ket{d_{xy}}\}$, is promoted from a static background to a fully dynamical degree of freedom. The effective Hamiltonian takes the form

$$\mathcal{H} = \varepsilon_0({\mathbf{k}}) \tau_0 \otimes \sigma_0 + \left[\bm{\tau} \cdot \mathbf{l}(\mathbf{r}, t)\right] \otimes \left[\varepsilon_\text{ani}(\mathbf{k}) \sigma_0 + J(\mathbf{k})\,\bm{\sigma} \cdot \mathbf{n}(\mathbf{r}, t)\right]$$

where $\bm{\tau}$ and $\bm{\sigma}$ are the orbital and spin Pauli matrices, and the specific forms of $\varepsilon_\text{ani}(\mathbf{k})$ and $J(\mathbf{k})$ encode the lattice-driven band anisotropy. Applying local gauge transformations yields emergent gauge potentials encoding the spatial and temporal gradients of both orbital and spin order parameters. Critically, the emergent electromagnetic fields split into additive orbital and spin components: $$\mathbb{E}_\text{em} = \mathbb{E}_l \otimes \sigma_0 + \tau_0 \otimes \mathbb{E}_s, \quad \mathbb{B}_\text{em} = \mathbb{B}_l \otimes \sigma_0 + \tau_0 \otimes \mathbb{B}_s$$ with generalized expressions involving triple products of the time and spatial derivatives of $l(\mathbf{r}, t)$0 or $l(\mathbf{r}, t)$1. Notably, these results generalize the classic Volovik/SMF/Topological Hall emergent field expressions to encompass arbitrary combinations of spin and orbital textures.

Orbital-Driven Transport Responses

Utilizing the extended Hamiltonian, the authors derive expressions for the emergent charge, spin, orbital, and magnetic octupole currents. The approach, based on the Drude model, considers the tensor structure of conductivities stemming from the band anisotropy factors $l(\mathbf{r}, t)$2 and $l(\mathbf{r}, t)$3. Transport is shown to be governed by both $l(\mathbf{r}, t)$4 (quantitative anisotropy corrections) and $l(\mathbf{r}, t)$5 (which vanishes for isotropic bands), with the operator $l(\mathbf{r}, t)$6 encoding the symmetry breaking of current directions.

Specifically:

• The emergent orbital electric field $l(\mathbf{r}, t)$7 and magnetic field $l(\mathbf{r}, t)$8 generate novel orbital and magnetic multipole currents, firmly establishing orbitals as a driver of non-trivial transport.
• Anisotropy-dependent currents ($l(\mathbf{r}, t)$9, $\{\ket{p_x}, \ket{p_y}\}$0) vanish in the isotropic limit, demonstrating the necessity of lattice-orbital coupling for the observation of these phenomena.
• Both charge and magnetic octupole currents can be independently controlled by the degree of band anisotropy and gate-tunable Fermi energy.
• The explicit expressions for Hall conductivities show how orbital N\'eel textures produce both topological orbital Hall and topological spin Hall effects, with the former being directly governed by orbital-derived EEMFs rather than spin-induced fields.

Lattice Distortion and Purely Orbital Emergent Transport

A critical and previously unaddressed mechanism emerges from the inclusion of dynamic lattice distortions, parameterized by a spatiotemporally varying distortion angle $\{\ket{p_x}, \ket{p_y}\}$1. Rotation of the orbital components by this angle induces additional EEMFs, $\{\ket{p_x}, \ket{p_y}\}$2 and $\{\ket{p_x}, \ket{p_y}\}$3, which are mathematically and physically distinct from those in undistorted systems.

Key results include:

• Lattice distortions alone, even without complex non-collinear textures, can yield nonvanishing EEMFs provided both time-dependent and spatially varying orbital configurations exist.
• The resulting distortion-induced magnetic octupole current survives even in the isotropic limit, in contrast with the pure spin and orbital cases, indicating a unique route for experimental detection and control.
• Proposed experimental architectures utilizing piezo-driven strain and optically induced orbital textures offer realistic verifiability of these purely orbital transport phenomena, distinct from thermoelectric and standard spintronic responses.

Implications and Future Directions

This work establishes that the orbital sector, when treated as a dynamic pseudospin, underlies a class of emergent transport phenomena unachievable in standard collinear (anti)ferromagnets, or when considering only the spin degree of freedom. The explicit link between band (lattice) anisotropy, orbital dynamics, and the emergence of electric and magnetic responses yields several directions for theoretical and experimental exploration:

• Extension to higher-order altermagnets: The orbital N\'eel formalism admits generalization to multipolar altermagnets, opening the possibility of inducing and detecting yet higher-rank magnetic multipole currents (e.g., hexadecapole, triakontadipole).
• Non-equilibrium orbitronics: The interplay between spin, orbital, and lattice degrees of freedom, especially away from equilibrium, suggests control paradigms for ultrafast electronic devices resilient to spin-orbit scattering effects.
• Reciprocal dynamical effects: Onsager relations imply that not only can strain drive orbital and magnetic multipole currents, but such currents may also induce lattice dynamics, potentially realizing current-driven phononic or structural phase transitions.
• Inclusion of spin-orbit coupling: Introduction of SOC would further entangle spin and orbital EEMFs, likely manifesting as nontrivial mixed-space Berry curvatures, as discussed in contexts such as mixed Weyl semimetals.

Conclusion

By systematically incorporating the dynamic orbital degree of freedom and treating lattice anisotropy as an active variable, this work demonstrates that altermagnets facilitate transport phenomena that are fundamentally unattainable in conventional systems. The formalism predicts robust, anisotropy-tunable orbital and magnetic multipole currents, including gate-controllable emergent responses driven by both orbital textures and lattice distortions. These results motivate the experimental realization of orbital-driven transport and multipole-based devices, with ramifications for future spintronics, orbitronics, and multipolar electronics. The approach provides an essential foundation for the ongoing reclassification of emergent transport in systems with complex magnetic symmetry.