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.