Ferromagnetic phase transitions of inhomogeneous systems modelled by square Ising models with diamond-type bond-decorations

The two-dimensional Ising model defined on square lattices with diamond-type bond-decorations is employed to study the nature of the ferromagnetic phase transitions of inhomogeneous systems. The model is studied analytically under the bond-renormalization scheme. For an n-level decorated lattice, the long-range ordering occurs at the critical temperature given by the fitting function (kBTc/J)n=1.6410+(0.6281)exp[−(0.5857)n], and the local ordering inside n-level decorated bonds occurs at the temperature given by the fitting function (kBTm/J)n=1.6410−(0.8063)exp[−(0.7144)n]. The critical amplitude Asing(n) of the logarithmic singularity in specific heat characterizes the width of the critical region, and it varies with the decoration-level n as Asing(n)=(0.2473)exp[−(0.3018)n], obtained by fitting the numerical results. The cross over from a finite-decorated system to an infinite-decorated system is not a smooth continuation. For the case of infinite decorations, the critical specific heat becomes a cusp with the height c(n)=0.639852. The results are compared with those obtained in the cell-decorated Ising model.