Thermodynamic Electric Toroidal Dipole and Intrinsic Longitudinal Spin Transport
Abstract
Electric toroidal dipoles (ETDs) characterize ferroaxial order, yet their bulk definition in periodic crystals has remained elusive because conventional multipole operators involve the ill-defined position operator. Here we formulate a thermodynamic ETD by coupling a spatially varying electric field to the relativistic spin-induced electric polarization. The resulting expression is gauge invariant and provides a bulk order parameter for ferroaxial phases. We further establish a direct relation between the chemical-potential derivative of the ETD and the intrinsic longitudinal spin conductivity in insulating systems. To demonstrate the formulation, we construct a minimal ferroaxial extension of the Kane--Mele model. The ETD becomes finite exclusively in the ferroaxial phase and is strongly enhanced near a small band gap, accompanied by a sizable longitudinal spin current. Our results establish a thermodynamic theory of ETDs in crystalline solids and identify the longitudinal spin conductivity as a direct transport manifestation of ferroaxial order.
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BibTeX
@misc{yamanaka2026thermodynamic,
title = {{Thermodynamic Electric Toroidal Dipole and Intrinsic Longitudinal Spin Transport}},
author = {Taisei Yamanaka and Takumi Sato and Satoru Hayami},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.02443},
eprint = {2609.02443},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.02443},
}
RIS
TY - UNPB
TI - Thermodynamic Electric Toroidal Dipole and Intrinsic Longitudinal Spin Transport
AU - Taisei Yamanaka
AU - Takumi Sato
AU - Satoru Hayami
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2609.02443
UR - https://arxiv.org/abs/2609.02443
AB - Electric toroidal dipoles (ETDs) characterize ferroaxial order, yet their bulk definition in periodic crystals has remained elusive because conventional multipole operators involve the ill-defined position operator. Here we formulate a thermodynamic ETD by coupling a spatially varying electric field to the relativistic spin-induced electric polarization. The resulting expression is gauge invariant and provides a bulk order parameter for ferroaxial phases. We further establish a direct relation between the chemical-potential derivative of the ETD and the intrinsic longitudinal spin conductivity in insulating systems. To demonstrate the formulation, we construct a minimal ferroaxial extension of the Kane--Mele model. The ETD becomes finite exclusively in the ferroaxial phase and is strongly enhanced near a small band gap, accompanied by a sizable longitudinal spin current. Our results establish a thermodynamic theory of ETDs in crystalline solids and identify the longitudinal spin conductivity as a direct transport manifestation of ferroaxial order.
ER -