Resolution of the Two-Dimensional Ferromagnetic Spin-3/2 Ising Model via Cluster Growth

We propose a computational methodology based on a hierarchical cluster growth process to solve spin-3/2 Ising models efficiently. The method circumvents the exponential complexity ($4^{N}$) of the canonical ensemble partition function by iteratively constructing finite magnetic clusters of size $N_g$, where the effective spin state of a site in generation $g+1$ is determined by the local magnetization of a cluster from generation $g$. This approach, which shares conceptual ground with effective field theories, allows the study of systems of effectively very large size $N = N_0 (N_g)^{g}$. We apply the formalism to the ferromagnetic spin-3/2 Ising model on a honeycomb lattice, modeling the monolayer CrI$_3$, a prototypical two-dimensional Ising magnet. The model, calibrated using the experimental transition temperature ($T_{c} \simeq 45$ K), successfully reproduces key experimental features: the temperature dependence of the magnetization $m(T)$, including its inflection point, and the broadened peak in the specific heat $c_v(T)$. We also compute the entropy $s(T)$, finding a finite residual value at low temperatures consistent with the system's double degeneracy. Our results demonstrate that this hierarchical cluster method provides a quantitatively accurate and computationally efficient framework for studying complex magnetic systems.