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

## TL;DR

Researchers created a nanomagnet structure that mimics the Blume-Capel model, allowing observation of exotic magnetic phases and transitions.

## Contribution

A Ruby lattice artificial spin ice is used to realize Blume-Capel degrees of freedom through toroidal moments.

## Key findings

- A two-step ordering process of toroidal moments was demonstrated in the Ruby artificial spin ice.
- The system exhibits a high-temperature crossover to an intermediate paratoroidic regime followed by a second-order transition to a ferrotoroidic state.
- The observed phases align with the Blume-Capel model's phase diagram, including a tricritical point.

## Abstract

Realizing
exotic Hamiltonians beyond the Ising model
is a key pursuit
in experimental statistical physics. One such 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 nanomagnets
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.