Fig. 2: Temperature- and size-dependent properties of isotropic 1D and 2D materials with different crystal structures.

a–d Comparative simulations of the temperature-dependent magnetisation for honeycomb, hexagonal, square lattices and an atomic chain (1D), respectively, for different system sizes. Points indicate the results of Monte Carlo simulations, the lines show fits to the Curie-Bloch Eq. (3) in the classical limit, and the shaded regions indicate the anisotropic spherical model calculations for different assumptions of the renormalisation factor for the Curie temperature arising from the mean-field approximation. See Supplementary Section 6 for details. The dashed and solid lines in d indicate the anisotropic spherical model calculations, and the exact solution, respectively. Both show a sound agreement with the atomistic simulations. The datasets in a–c clearly show the existence of short-range collinear magnetic order for all 2D lattices at the simulated sizes considered with nonzero crossover temperature. Zero magnetic anisotropy is included in all calculations.