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

TL;DR

This work tackles the computational challenge of simulating a spin-3/2 Ising model on a 2D honeycomb lattice by introducing a hierarchical cluster-growth framework. The method builds generations of clusters with local magnetizations, replacing the full $4^{N}$ state space with a scale-dependent, coarse-grained description and an effective coupling that evolves as $K^{(g)} = \gamma^{(0)} a^{g}/T$, enabling the study of very large domains. Applied to CrI$_3$, the approach reproduces key thermodynamic features: a magnetization curve $m(T)$ with an inflection near the specific-heat peak, a pronounced $c_v(T)$ maximum near $T_c \approx 45$ K, and a finite residual entropy from ground-state degeneracy, with critical-exponent estimates suggesting a departure from mean-field behavior and convergence toward 2D Ising universality as domain size grows. The results demonstrate a quantitatively accurate, computationally efficient framework for analyzing complex 2D magnetic systems and their critical properties, while preserving a clear EFT/RG-inspired interpretation of scale evolution.

Abstract

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.

Figures (12)

• Figure 1: In (a) the first-generation cluster $g=1$ is illustrated, where each $m_j$ corresponds to a cluster mean. In (b) is the zero-generation cluster is depicted.
• Figure 2: Magnetization versus temperature for $g=0$ and for diferent values of the parameter $a$, where $g=10$.
• Figure 3: Scale evolution of the magnetization $m^{(g)}(T;a)$ as a function of the generation index $g$, obtained from the canonical partition sum of the cluster. Results are shown for three representative temperatures ($T=0.8T_c$, $T=T_c$,and $T=1.2T_c$) and for different values of the scale parameter $a$. Note that here we have $g_{max}=31$.
• Figure 4: Internal Energy versus $a$ for the volumetric scales of the magnetic domains adopted in the modeling.
• Figure 5: The behavior of the derivative $u\prime (a)$ is shown in order to determine the minima of these derivatives. The case for the nanometric volume is shown in the inset.