Fig. 2: Temperature dependence of the unconventional ferromagnetic state.

a Graphical solution of the self-consistency equation for the mean value of spin x component, \(\left\langle {S}_{x}\right\rangle /S=S{B}_{S}\left(4\beta {\Lambda }_{x}{\langle {S}_{x}\rangle }^{3}/{S}^{3}\right)\), for various coupling strengths and S ≫ 1. A solution is obtained when the solid line crosses the dashed line y = 〈S_x〉. k_B and T are the Boltzmann constant and temperature, respectively. b Temperature dependence of the absolute mean value of the x component of the spin for an array of cylindrical (black line) and spherical (orange line) nanoparticles, with solid (dotted) lines indicating locally stable (unstable) solutions. The thick solid lines indicate that the solution furthermore corresponds to the global minimum of the free energy. c Dependence of the critical temperature T_c on the lattice constant a of the square (solid line) and hexagonal (dashed line) array of spherical (orange line) or cylindrical (black line) nanoparticles.