  Metallic discs engineered on the 100 nm scale have an internal magnetic
texture which varies from a fully magnetized state to a vortex state with zero
moment. The interplay between this internal structure and the inter-disc
interactions is studied in magnetic metamaterials made of square arrays of the
magnetic discs. The texture is modeled by a mesospin of varying length with
O(2) symmetry and the inter-disc interaction by a nearest neighbour coupling
between mesospins. The thermodynamic properties of the model are studied
numerically and an ordering transition is found which varies from
Kosterlitz-Thouless to first order via an apparent tri-critical point. The
effective critical exponent characterising the finite size magnetization
evolves from the value for the 2D-XY model to less than half this value at the
tri-critical point. The consequences for future experiments both in and out of
equilibrium are discussed.
