Long-range exchange coupling in a magnetic multilayer system

In this study, we explored the magnetic coupling in a multilayer system consisting of thin layers separated by a distance $d$. We have employed Monte Carlo (MC) simulations to calculate the thermodynamic quantities such as the magnetization per spin $m_{L}^{\mu}$, magnetic susceptibility $\chi_{L}^{\mu}$, and the reduced fourth-order Binder cumulant $U_{L}^{\mu}$ as a function of temperature $T$ and for several values of lattice size $L$. These quantities were obtained for each layer ($\mu=l)$, for the bulk system ($\mu=b)$, as a function of interlayer distance $d$ and the parameter $\alpha$ that defines the scale length of the exponential decay of the interaction. Furthermore, we applied the finite-size scaling theory to calculate the critical exponents. Our results reveal that the system exhibits 2D Ising exponents when the layers are sufficiently separated. On the other hand, in the compact limit, where $d$ equals the distance between two adjacent sites within the same layer, our bulk results show that the system exhibits 3D Ising critical exponents, provided that the number of layers $L_{z}$ increases proportionally to the layer sizes. Even with the separation increasing up to $d=2.0$, the layers are so correlated that the set of critical exponents retains the values of 3D critical exponents. However, when the number of layers $L_{z}$ remains fixed, even in the compact limit with periodic boundary conditions, only the exponent $\beta$ aligns closely with the predicted literature values, on the other hand, the other exponents show significant deviations.