Realizing Blume-Capel Degrees of Freedom with Toroidal Moments in a Ruby Artificial Spin Ice

Realizing exotic Hamiltonians beyond the Ising model is a key pursuit in experimental statistical physics. Onesuch example is the Blume-Capel model, a three-state spin model, whose phase diagram features a tricritical point where second-order and first-order transition lines converge, leading to a coexistence of paramagnetic, ferromagnetic, and disordered phases. Here, we realize an artificial crystal of single-domain nanomagnets, placed on the links of the Ruby lattice, enabling real-space observation of the Blume-Capel degrees of freedom. 

These Blume-Capel degrees of freedom are represented by the presence, sign and interactions of the toroidal moments that emerge naturally in plaquettes of nano-magnets in the Ruby artificial spin ice. By precisely tuning the lattice parameters of the Ruby artificial spin ice, we demonstrate control over the two-step ordering process of the toroidal moments, whereby there is a high-temperature crossover from a paramagnetic phase to an intermediate paratoroidic regime, followed by a second-order phase transition to a ferrotoroidic ground state. This sequence of toroidal phases and transitions is accurately captured by the Blume-Capel framework and provides a direct realization of a substantial portion of the phase diagram associated with the model. This establishes a platform for exploring exotic Hamiltonians in terms of artificial spin ice superstructures, here with groups of nanomagnets forming toroidal moments. The success of this mapping highlights the potential of intentionally engineered lattice designs, whose effective Hamiltonians can mediate unconventional forms of magnetic order with distinct behaviors and functionalities.