We theoretically show that the critical temperature T c of the critical phenomena described within the Gibbs distribu- tion is varying linearly with the energetic parameters of the Hamiltonian. This kind of variation is due to the nature of the competition between the energy of the system and the energy kBT the thermic source. The change of this competition will lead to a different variation of the critical temperature. This linear variation has been reported by well-known theories of the simplest Ising model. In this study, we observe that this linearity concerns all well- known magnetic Hamiltonians formulated through spin-spin interactions. This competition has also a direct effect on the class of universalities in critical phenomena.
We theoretically show that the critical temperature T c of the critical phenomena described within the Gibbs distribu- tion is varying linearly with the energetic parameters of the Hamiltonian. This kind of variation is due to the nature of the competition between the energy of the system and the energy kBT the thermic source. The change of this competition will lead to a different variation of the critical temperature. This linear variation has been reported by well-known theories of the simplest Ising model. In this study, we observe that this linearity concerns all well- known magnetic Hamiltonians formulated through spin-spin interactions. This competition has also a direct effect on the class of universalities in critical phenomena.
Critical behavior of Ising ferromagnetic systems with the competition between short- and long-range interactions near the point of a second-order phase transition has been investigated. The effects of long-range interaction lead to the need to calculate the influence of next-nearest spins, which lie inside the sphere of a specific radius. The study was carried out for a power law of decreaseing the exchange integral with distance for a three-dimensional Ising model with a cubic lattice. A comparison with the results of a field-theoretical approach and data of real experiments has been carried out. It has been shown that the critical temperature grows linearly with an increase in the parameters of long-range interaction. Critical exponents have been calculated at different parameters of long-range interaction. It is shown that at small dimensions of a system it is the short-range interaction that exerts the dominant effect on the critical behavior of the system. With an increase in the linear dimension of the system, long-range forces begin to dominate.
The standard Theory of Critical Phenomena within Boltzmann-Gibbs statistical mechanics is known to successfully predict, at any temperature differing from a continuous-phase-transition critical one, the (finite) values for thermodynamically relevant quantities such as isothermal magnetic susceptibility, specific heat, correlation length, Grüneisen parameter, and similar ones. However, at the precise critical point, empirically unreachable infinities emerge within that approach at the N→∞ thermodynamical limit. Such divergences, although mathematically correct, may be seen as poorly informative ones. Indeed, they make no distinction at all between say the Ising, XY, Heisenberg, and other models. Moreover, for say the XY model, they make no distinction whether the spins are, for instance, 1/2, 1 or 3/2, or whether the interactions are say first-neighbor or first-and-second-neighbor ones. The introduction of nonadditive entropic functionals within a consistently generalized statistical mechanics substantially improves the situation, i.e., it regularizes the theory providing informative finite values instead of mere infinities. This undoubtedly enriching fact is illustrated [Soares et al., Phys. Rev. B 111, L060409 (2025)] for the Grüneisen parameter at the zero-temperature quantum critical point of a spin-1/2 ferromagnetic Ising chain in the presence of a transverse magnetic field, by introducing the nonadditive entropic functional Sq with the unique value q=q⋆≡37-6=0.0828…, which precisely guarantees the thermodynamical extensivity of the entropy as mandated by the preservation of the Legendre-transform structure of classical thermodynamics.
The critical behavior of the disordered ferromagnetic Ising model is studied numerically by the Monte Carlo method in a wide range of variation of concentration of nonmagnetic impurity atoms. The temperature dependences of correlation length and magnetic susceptibility are determined for samples with various spin concentrations and various linear sizes. The finite-size scaling technique is used for obtaining scaling functions for these quantities, which exhibit a universal behavior in the critical region; the critical temperatures and static critical exponents are also determined using scaling corrections. On the basis of variation of the scaling functions and values of critical exponents upon a change in the concentration, the conclusion is drawn concerning the existence of two universal classes of the critical behavior of the diluted Ising model with different characteristics for weakly and strongly disordered systems.
We consider the critical system with a point defect and study the variation of thermodynamic quantities, which are the differences between those with and without the defect. Within renormalization group theory, we show generally that the critical exponent of the internal energy variation is the specific heat exponent of a pure system, and the critical exponent of the heat capacity variation is that for the temperature derivative of specific heat of a pure system. This conclusion is valid for the isotropic systems with a short-range interaction. As an example we solve the two-dimensional Ising model with a point defect numerically. The variations of the free energy, internal energy, and specific heat are calculated with the bond propagation algorithm. At the critical point, the internal energy variation diverges with the lattice size logarithmically and the heat capacity variation diverges with size linearly. Near the critical point, the internal energy variation behaves as ln|t| and the heat capacity variation behaves as |t|^{-1}, where t is the reduced temperature.
Order parameter fluctuations for the two-dimensional Ising model in the region of the critical temperature are presented. A locus of temperatures T(*) (L) and a locus of magnetic fields B(*) (L) are identified, for which the probability density function is similar to that for the two-dimensional XY model in the spin wave approximation. The characteristics of the fluctuations along these points are largely independent of universality class. We show that the largest range of fluctuations relative to the variance of the distribution occurs along these loci of points, rather than at the critical temperature itself and we discuss this observation in terms of intermittency. Our motivation is the identification of a generic form for fluctuations in correlated systems in accordance with recent experimental and numerical observations. We conclude that a universality-class-dependent form for the fluctuations is a particularity of critical phenomena related to the change in symmetry at a phase transition.
Order parameter fluctuations for the two dimensional Ising model in the region of the critical temperature are presented. A locus of temperatures T*(L) and of magnetic fields B*(L) are identified, for which the probability density function is similar to that for the 2D-XY model in the spin wave approximation.The characteristics of the fluctuations along these points are largely independent of universality class. We show that the largest range of fluctuations relative to the variance of the distribution occurs along these loci of points, rather than at the critical temperature itself and we discuss this observation in terms of intermittency. Our motivation is the identification of a generic form for fluctuations in correlated systems in accordance with recent experimental and numerical observations. We conclude that a universality class dependent form for the fluctuations is a particularity of critical phenomena related to the change in symmetry at a phase transition.