Comparative study of magnetic properties of Mn$^{3+}$ magnetic clusters in GaN using classical and quantum mechanical approach

Currently, simulations of many-body quantum systems are known to be computationally too demanding to be solved on classical computers. The main problem is that the computation time and memory necessary for performing the calculations usually grow exponentially with the number of particles $N$. An efficient approach to simulate many-body quantum systems is the use of classical approximation. However, it is known that at least at low temperatures, the allowed spin fluctuations in this approach are overestimated what results in enhanced thermal fluctuations. It is therefore timely and important to assess the validity of the classical approximation. To this end, in this work, we compare the results of numerical calculations of small Mn$^{3+}$ magnetic clusters in GaN, where the Mn spins are treated classically with those where they are treated quantum-mechanically (crystal field model). In the first case, we solve the Landau-Lifshitz-Gilbert (LLG) equation that describes the precessional dynamics of spins represented by classical vectors. On the other hand, in the crystal field model, the state of Mn$^{3+}$ ion ($d^4$ configuration with $S=2$, $L=2$) is characterized by the set of orbital and spin quantum numbers $|m_s,m_L>$. Particular attention is paid to use numerical parameters that ensure the same single ion magnetic anisotropy in both classical and quantum approximation. Finally, a detailed comparative study of magnetization $\mathbf{M}(\mathbf{H}, T)$ as a function of the magnetic field $\mathbf{H}$, temperature $T$, number of ions in a given cluster $N$ and the strength of super-exchange interaction $J$, obtained from both approaches will be presented.