Finite-Temperature Toroidal Moment Amenable to Direct Observation in an Fe$_{10}$Dy$_{10}$ Molecular Ring

Abstract

Single-molecule toroics (SMTs) host closed magnetic-vortex configurations that carry toroidal moments $\boldsymbol{\tau}$, whose electric-dipole symmetry enables magnetoelectric spin control. Yet, opposite toroidal chiralities are degenerate in conventional magnetic fields, making direct detection of molecular toroidal polarisation challenging. Current approaches probe molecular toroidal dynamics only indirectly through weak residual magnetism, leaving direct interrogation of toroidal polarisation an open challenge. Moreover, the survival of toroidal polarization at finite temperature, and realistic preparation-and-readout conditions, have not been quantitatively established. Here we investigate the icosanuclear $3d$--$4f$ molecular ring Fe$_{10}$Dy$_{10}$, featuring a $\sim$62-billion-dimensional low-energy manifold with pervasive toroidal character, rendered computationally tractable via an ab initio-informed transfer-matrix framework with perturbative corrections. Our model reproduces magnetic and calorimetric measurements and reveals a maximally toroidal ground doublet with robust finite-temperature toroidal response. We introduce the toroidal susceptibility $\xi$ as a finite-temperature linear-response function to quantify toroidal polarisation induced by magnetic-field curl. We then develop a preparation-and-detection protocol in which a temporally asymmetric near-infrared waveform generates a cumulative toroidal population imbalance, while an ab initio-informed magnetoelectric tensor predicts an electric-field-induced magnetic moment within $\mu$SQUID detectability. These results establish Fe$_{10}$Dy$_{10}$ as a molecular platform where toroidal polarisation can be prepared, accumulated and read out under realistic experimental conditions.