Abstract This study investigates the critical properties of the mean-field Ising model incorporating five-spin interactions and external magnetic field. By using a variational approach to minimize free energy, the behaviour of the magnetic order parameter is systematically analysed across different phases. The research establishes the conditions for phase transitions, identifies critical points, and calculates the associated critical exponents. The model demonstrates unique phase behaviours, including the coexistence of thermodynamic equilibria and smooth transitions in magnetization. These results offer valuable insights into the impact of higher-order spin interactions in magnetic systems, with implications for understanding complex multi-spin dynamics in physical and biological networks.
In the framework of mean field approximation, we consider a spin system consisting of two interacting sub-ensembles. The intra-ensemble interactions are ferromagnetic, while the inter-ensemble interactions are antiferromagnetic. We define the effective number of the nearest neighbors and show that if the two sub-ensembles have the same effective number of the nearest neighbors, the classical form of critical exponents (α=0, β=1/2, γ=γ'=1, δ=3) gives way to the non-classical form (α=0, β=3/2, γ=γ'=0, δ=1), and the scaling function changes simultaneously. We demonstrate that this system allows for two second-order phase transitions and two first-order phase transitions. We observe that an external magnetic field does not destroy the phase transitions but only shifts their critical points, allowing for control of the system's parameters. We discuss the regime when the magnetization as a function of the magnetic field develops a low-magnetization plateau and show that the height of this plateau abruptly rises to the value of one when the magnetic field reaches a critical value. Our analytical results are supported by a Monte Carlo simulation of a three-dimensional layered model.
Analytical expressions of the order parameters near the transition temperatures in the spin-3/2 Ising system with bilinear (J) and biquadratic (K) interactions are presented for various values of J/K. First, we obtain the free energy expression and the equations to determine order parameters by using the mean-field approximation. Then, the order parameters are expressed in the vicinity of the transition temperatures in which these expressions are very important to study the dynamics of the system by means of Onsager's theory of irreversible thermodynamics. Hence, we investigate the phase transitions occurring in the system and also obtain two tricritical points analytically. Finally, the specific heat and magnetic susceptibility are calculated and an argument about the critical exponents at the second-order phase transitions and tricritical points is given.
Abstract The effect of an external magnetic field on the phase transitions and the thermodynamic and magnetic properties of the three-dimensional antiferromagnetic Ising model on a body-centered cubic lattice has been studied using a replica algorithm of the Monte Carlo method, taking into account the interactions of next nearest neighbors. A phase diagram of the dependence of the critical temperature on an external magnetic field has been built. It is shown that a second-order phase transition is observed in the magnetic field range under consideration.
Universality classes encompass the analogous thermodynamic behavior of unlike physical systems, at different spatial dimensions
d, in the vicinity of their critical point. Critical exponents define these classes, with the Ising model being the outstanding prototype that elucidates the differences from the mean-field category, believed to be valid above a critical dimension only. Here, in apparent striking contradiction to the Ising universality class, we demonstrate that the critical behavior of a finite Ising system of
N spins in
d=3 obeys mean-field Landau theory in the vicinity of its critical point, with classical critical exponents. Yet, when expressed in terms of the linear size
L of the system, the free energy unveils its proper finite-size scaling form, from which the thermodynamic limit critical temperature
Tc and the Ising critical exponents
ν,
γ and
β can be identified. We find that the larger the size
L, the smaller the mean-field region, shrinking to zero in the thermodynamic limit. These conclusions are achieved via the use of an alternative approach to collect data from a Monte Carlo simulation of a three-dimensional Ising model that allows for the evaluation of the free energy per spin
f=f(T,m;L) and of the coexistence curve, or spontaneous magnetization at zero magnetic field,
mcoex=m(T;L) as functions of temperature
T and magnetization per spin
m=M/N. Our results suggest a revision of the role of mean-field theory in the elucidation of critical phenomena.