Quantum Toroidicity in Single-Molecule Toroics: A Unifying Model Based on Heisenberg Spin Rings

In this review we explore the electronic structure of Heisenberg quantum spin rings and identify various microscopic ingredients that are necessary to achieve a degenerate ground state supporting a non-zero toroidal moment, the hallmark of Single-Molecule Toroics (SMTs). Our analysis goes beyond the semi-classical, non-collinear Ising model typically used to describe lanthanide-based SMTs, thus accounting for quantum tunnelling of toroidal states in SMTs with and without strong, on-site, magnetic anisotropy. Non-collinear Ising SMTs are recovered by our model as the limiting case of a (pseudo) spin Heisenberg ring with infinitely large on-site easy-axis anisotropy. In the strong but finite anisotropy limit, we show how isotropic exchange works to disrupt the formation of toroidal ground states, resulting in the quantum tunnelling of the toroidal moment, and we expose a non-intuitive geometric dependence of the toroidal quantum tunnelling rates on the arrangement of the local magnetic axes. In the opposite strong exchange and negligible on-site magnetic anisotropy limit, we show that SMTs can still be achieved in antiferromagnetically coupled odd-membered Heisenberg rings, as consequence of the extra-degeneracy associated with spin frustration. While only the triangular geometry gives rise to a SMT, with larger odd-membered rings featuring toroidal moments only in excited multiplets, we find that next-nearest neighbour exchange coupling, intramolecular magnetic dipolar coupling and Dzyaloshinskii–Moriya antisymmetric coupling can in some limit recover toroidal moments in the ground state, opening up a route to achieve SMTs in Heisenberg quantum rings of arbitrary size with negligible on-site anisotropy. Finally, we explore the incorporation of a hetero-atom into the Heisenberg ring and illustrate analytically how this extra degree of freedom can be utilised to engineer heterometallic triangles with more robust toroidal ground states for use in future SMT devices.