New Random Ordered Phase in Isotropic Models with Many-body Interactions

In this study, we have found a new random ordered phase in isotropic models with many-body interactions. Spin correlations between neighboring planes are rigorously shown to form a long-range order, namely coplanar order, using a unitary transformation, and the phase transition of this new order has been analyzed on the bases of the mean-field theory and correlation identities. In the systems with regular 4-body interactions, the transition temperature $T_{\text{c}}$ is obtained as $T_{\text{c}}=(z-2)J/k_{\text{B}}$, and the field conjugate to this new order parameter is found to be $H^2$. In contrast, the corresponding physical quantities in the systems with random 4-body interactions are given by $T_{\text{c}}=\sqrt{z-2}J/k_{\text{B}}$ and $H^4$, respectively. Scaling forms of order parameters for regular or random 4-body interactions are expressed by the same scaling functions in the systems with regular or random 2-body interactions, respectively. Furthermore, we have obtained the nonlinear susceptibilities in the regular and random systems, where the coefficient $\chi_{\text{nl}}$ of $H^3$ in the magnetization shows positive divergence in the regular model, while the coefficient $\chi_{7}$ of $H^7$ in the magnetization shows negative divergence in the random model.