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 as $(k_{B}T_{c}/J)_{n}=1.6410+(0.6281) \exp [ -(0.5857) n]$, and the local ordering inside $n$-level decorated bonds occurs at the temperature given by the fitting function as $(k_{B}T_{m}/J)_{n}=1.6410-(0.8063) \exp [ -(0.7144) n]$. The critical amplitude $A_{\sin g}^{(n)}$ of the logrithmic singularity in specific heat characterizes the width of the critical region, and it varies with the decoration level $n$ as $A_{\sin g}^{(n)}=(0.2473) \exp [ -(0.3018) n]$, 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.