In this paper we analytically study the recurrence equations of an Ising model with three competing interactions on a Cayley tree of order three. We exactly describe paramagnetic and ferromagnetic phases of the Ising model. We obtain some rigorous results: critical temperatures and curves, number of phases, partition function. Ganikhodjaev et al. [J. Concrete and Applicable Mathematics, 9 (1), 26-34 (2011)] have numerically studied the Ising model on a second-order Cayley tree. We compare the numerical results to exact solutions of mentioned model.
In the present paper, the Ising model with mixed spin-(1,1/2) is considered on the second order Cayley tree. A construction of splitting Gibbs measures corresponding the model is given which allows to establish the existence of the phase transition (non-uniqueness of Gibbs measures). We point out that, in the phase transition region, the considered model has three translation-invariant Gibbs measures in the ferromagnetic and anti-ferromagnetic regimes, while the classical Ising model does not possesses such Gibbs measures in the anti-ferromagnetic regime. It turns out that the considered model…
We consider the ferromagnetic Ising model with spatially dependent external fields on a Cayley tree, and we investigate the conditions for the existence of the phase transition for a class of external fields, asymptotically approaching a homogeneous critical external field. Our results extend earlier results by Rozikov and Ganikhodjaev.
In this present paper, the recurrence equations of an Ising model with three coupling constants on a third-order Cayley tree are obtained. Paramagnetic and ferromagnetic phases associated with the Ising model are characterized. Types of phases and partition functions corresponding to the model are rigorously studied. Exact solutions of the mentioned model are compared with the numerical results given in Ganikhodjaev et al. [J. Concr. Appl. Math., 2011, 9, No. 1, 26-34].
Some results on the two-point function and on the analytic structure of the momenta of the effective fugacity at the origin for a class of diluted ferromagnetic Ising models on the Cayley tree are presented.
The main purpose of this review paper is to give systematically all the known results on phase diagrams corresponding to lattice models (Ising and Potts) on Cayley tree (or Bethe lattice) and chandelier networks. A detailed survey of various modelling applications of lattice models is reported. By using Vannimenus's approach, the recursive equations of Ising and Potts models associated to a given Hamiltonian on the Cayley tree are presented and analyzed. The corresponding phase diagrams with programming codes in different programming languages are plotted. To detect the phase transitions in th…
It is found that the zero field susceptibility chi of the Ising model on the Cayley tree exhibits unusually weak divergence at the critical point Tc. The susceptibility amplitude is found to diverge at Tc proportionally to the tree generation level n, while the behavior of chi is otherwise analytic in the vicinity of Tc, with the critical exponent gamma=0.
An exact analytical derivation is presented, showing that the Ising model on the Cayley tree exhibits a line of third order phase transition points, between temperatures $T_2=2k_B^{-1}J\ln ({\sqrt 2}+1) $ and $T_{BP}=k_B^{-1}J\ln (3)$, and a line of fourth order phase transitions between $T_{BP}$ and $\infty$, where $k_B$ is the Boltzmann constant, and $J$ is the nearest-neighbor interaction parameter.
While the Ising model on the Cayley tree has no spontaneous magnetization at nonzero temperatures in the thermodynamic limit, we show that finite systems of astronomical sizes remain magnetically ordered in a wide temperature range, if the symmetry is broken by fixing an arbitrary single (bulk or surface) spin. We compare the behavior of the finite size magnetization of this model with that of the Ising model on both the Sierpinski Gasket, and the one-dimensional linear chain. This comparison reveals the analogy of the behavior of the present model with the Sierpinski Gasket case.
A general analytical formula for recurrence relations of multisite interaction Ising models in an external magnetic field on the Cayley-type lattices is derived. Using the theory of complex analytical dynamics on the Riemann sphere, a numerical algorithm to obtain Yang-Lee and Fisher zeros of the models is developed. It is shown that the sets of Yang-Lee and Fisher zeros are almost always fractals, that could be associated with Mandelbrot-like sets on the complex magnetic field and temperature planes respectively.
The main aim of the present paper is to prove the existence of a phase transition in quantum Markov chain (QMC) scheme for the Ising type models on a Cayley tree. Note that this kind of models do not have one-dimensional analogous, i.e. the considered model persists only on trees. In this paper, we provide a more general construction of forward QMC. In that construction, a QMC is defined as a weak limit of finite volume states with boundary conditions, i.e. QMC depends on the boundary conditions. Our main result states the existence of a phase transition for the Ising model with competing inte…
We study the ground-state (T = 0) morphologies in the d = 3 random-field Ising model (RFIM) using a computationally efficient graph-cut method. We focus on paramagnetic states which arise for disorder strengths Δ> Δc, where Δc is the critical disorder strength at T = 0. These paramagnetic states consist of correlated "domains" of up and down spins which are separated by rough, fractal interfaces. They show novel scattering properties with a cusp singularity in the correlation function at short distances.
We consider the non-conserved dynamics of the Ising model on the two-dimensional square lattice, where each spin is influenced preferentially by its East and North neighbours. The single-spin flip rates are such that the stationary state is Gibbsian with respect to the usual ferromagnetic Ising Hamiltonian. We show the existence, in the paramagnetic phase, of a dynamical transition between two regimes of violation of the fluctuation-dissipation theorem in the nonequilibrium stationary state: a regime of weak violation where the stationary fluctuation-dissipation ratio is finite, when the asymm…
We study the $\pm J$ three-dimensional Ising model with a spatially uniaxially anisotropic bond randomness on the simple cubic lattice. The $\pm J$ random exchange is applied in the $xy$ planes, whereas in the z direction only a ferromagnetic exchange is used. After sketching the phase diagram and comparing it with the corresponding isotropic case, the system is studied, at the ferromagnetic-paramagnetic transition line, using parallel tempering and a convenient concentration of antiferromagnetic bonds ($p_z=0 ; p_{xy}=0.176$). The numerical data point out clearly to a second-order ferromagnet…
In this Monte Carlo study we concentrated on the influence of non-magnetic impurities arranged as the lines with random orientation on paramagnetic-to-ferromagnetic phase transition in the 3D Ising model. Special emphasize is given to the long-distance decay of the impurity-impurity pair correlation function. It is shown that for the lattice sizes considered (L=10-96) and for the two different impurity distributions (purely random and mutually avoiding lines) the function is governed by the power law of 1/r^a with an universal exponent a \simeq 2. This result supports our findings about the nu…
In this paper, we consider Vannimenus model with competing nearest-neighbors and prolonged next-nearest-neighbors interactions on a Cayley tree. For this model we define Markov random fields with memory of length 2. By using a new approach, we obtain new sets of Gibbs measures of Ising-Vannimenus model on Cayley tree of order 2. We construct the recurrence equations corresponding Ising-Vannimenus model. We prove the Kolmogorov consistency condition. We investigate the translation-invariant and periodic non transition-invariant Gibbs measures for the model. We find new sets of Gibbs measures di…
A coupled spin-electron diamond chain with localized Ising spins placed on its nodal sites and mobile electrons delocalized over interstitial sites is explored in a magnetic field taking into account the difference between Landé g-factors of the localized spins and mobile electrons. The ground-state phase diagram is constituted by two classical ferrimagnetic phases, the quantum unsaturated paramagnetic phase and the saturated paramagnetic phase. Both classical ferrimagnetic phases as well as the unsaturated paramagnetic phase are reflected in a low-temperature magnetization curve as intermedia…
The phase diagram of the random field Ising model on the Bethe lattice with a symmetric dichotomous random field is closely investigated with respect to the transition between the ferromagnetic and paramagnetic regime. Refining arguments of Bleher, Ruiz and Zagrebnov [J. Stat. Phys. 93, 33 (1998)] an exact upper bound for the existence of a unique paramagnetic phase is found which considerably improves the earlier results. Several numerical estimates of transition lines between a ferromagnetic and a paramagnetic regime are presented. The obtained results do not coincide with a lower bound for …
An axial next-nearest-neighbor Ising (ANNNI) model is studied by using the non-equilibrium relaxation method. We find that the incommensurate stripe phase between the ordered phase and the paramagnetic phase is negligibly narrow or may vanish in the thermodynamic limit. The phase transition is the second-order transition if approached from the ordered phase, and it is of the Kosterlitz-Thouless type if approached from the paramagnetic phase. Both transition temperatures coincide with each other within the numerical errors. The incommensurate phase which has been observed previously is a parama…
We study the out-of-time-order correlation (OTOC) of the Floquet transverse Ising model and use it to verify the phase diagram of the system. First, we present the exact analytical solution of the transverse magnetization OTOC using the Jorden-Wigner transformation. We calculate the speed of correlation propagation and analyze the behavior of the revival time with the separation between the observables. In order to get the phase structure of the Floquet transverse Ising system, we use the longitudinal magnetization OTOC as it is known to serve as an order parameter of the system. We show the p…
Highly frustrated antiferromagnets composed of magnetic rare-earth moments are currently attracting much experimental and theoretical interest. Rare-earth ions generally have small exchange interactions and large magnetic moments. This makes it necessary to understand in detail the role of long-range magnetic dipole-dipole interactions in these systems, in particular in the context of spin-spin correlations that develop in the paramagnetic phase, but are often unable to condense into a conventional long-range magnetic ordered phase. This scenario is most dramatically emphasized in the frustrat…
We examine the low frequency spin susceptibility of the paramagnetic phase of the quantum Ising chain in transverse field at temperatures well below the energy gap. We find that the imaginary part is dominated by rare quantum processes in which the number of quasiparticles changes by an odd number. We obtain exact results for the NMR relaxation rate in the low temperature limit for the integrable model with nearest-neighbor Ising interactions, and derive exact universal scaling results applicable to generic Ising chains near the quantum critical point. These results resolve certain discrepanci…
Critical neutron scattering measurements have been performed on CsCo$_{0.83}$Mg$_{0.17}$Br$_3$, a dilute stacked triangular lattice (STL) Ising antiferromagnet (AF). A two component lineshape associated with the critical fluctuations appears at a temperature coincident with T$_{N1}$ observed in pure CsCoBr$_3$. Such scattering is indicative of fluctuations in prototypical random field Ising model (RFIM) systems. The random field domain state arises in this case due to geometrical frustration within the STL Ising AF, which gives rise to a 3 sublattice Neel state, in which one sublattice is diso…
Zeros of the moment of the partition function $[Z^n]_{\bm{J}}$ with respect to complex $n$ are investigated in the zero temperature limit $β\to \infty$, $n\to 0$ keeping $y=βn \approx O(1)$. We numerically investigate the zeros of the $\pm J$ Ising spin glass models on several Cayley trees and hierarchical lattices and compare those results. In both lattices, the calculations are carried out with feasible computational costs by using recursion relations originated from the structures of those lattices. The results for Cayley trees show that a sequence of the zeros approaches the real axis of $…
In this paper, we provide the exact expression for the coefficients in the low-temperature series expansion of the partition function of the two-dimensional Ising model on the infinite square lattice. This is equivalent to exact determination of the number of spin configurations at a given energy. With these coefficients, we show that the ferromagnetic--to--paramagnetic phase transition in the square lattice Ising model can be explained through equivalence between the model and the perfect gas of energy clusters model, in which the passage through the critical point is related to the complete …
The phase transitions occurring in the frustrated Ising square antiferromagnet with first- ($J_1 < 0$) and second- ($J_2 < 0$) nearest-neighbor interactions are studied within the framework of the effective-field theory with correlations based on the different cluster sizes and for a wide range of $R = J_2/|J_1|$. Despite the simplicity of the model, it has proved difficult to precisely determine the order of the phase transitions. In contrast to the previous effective-field study, we have found a first-order transition line in the region close to $R = -0.5$ not only between the superantiferro…
We provide a concrete and systematic connection between the statistical physics of the Ising ferromagnet on a Cayley tree, and the study of memory in exponentially expanding spaces. Memory turns out to be a clear signal of the `Bethe-Peierls' phase transition, and the average of memory divided by its standard deviation provides a clear signal of the `spin-glass' transition temperature. Numerical Monte Carlo simulations are used to make transparent the existence of the two different transition temperatures. The quantities used to spot the phase transitions with Monte Carlo could be useful when …
This paper is devoted to an in-depth study of the limiting measure of Lee--Yang zeroes for the Ising Model on the Cayley Tree. We build on previous works of Müller-Hartmann-Zittartz (1974 and 1977), Barata--Marchetti (1997), and Barata--Goldbaum (2001), to determine the support of the limiting measure, prove that the limiting measure is not absolutely continuous with respect to Lebesgue measure, and determine the pointwise dimension of the measure at Lebesgue a.e. point on the unit circle and every temperature. The latter is related to the critical exponents for the phase transitions in the mo…
We investigate the phase diagram of a spin--1 Ising spin-glass model on a Cayley tree. According to early work of Thompson and collaborators, this problem can be formulated in terms of a set of nonlinear discrete recursion relations along the branches of the tree. Physically relevant solutions correspond to the attractors of these mapping equations. In the limit of infinite coordination of the tree, and for some choices of the model parameters, we make contact with findings for the phase diagram of more recently investigated versions of the Blume-Emery-Griffiths spin-glass model. In addition t…
The equilibrium phase behavior of microphase-forming systems is notoriously difficult to obtain because of the extended metastability of their modulated phases. In this paper we present a systematic simulation methodology for studying layered microphases and apply the approach to two prototypical lattice-based systems: the three-dimensional axial next-nearest-neighbor Ising (ANNNI) and Ising-Coulomb (IC) models. The method involves thermodynamically integrating along a reversible path established between a reference system of free spins under an ordering field and the system of interest. The r…
Mixed spin-1/2 and spin-1 Ising ferrimagnets on a triangular lattice with sublattices A, B and C are studied for two spin value distributions $(S_{\rm A},S_{\rm B},S_{\rm C})=(1/2,1/2,1)$ and $(1/2,1,1)$ by Monte Carlo simulations. The non-bipartite character of the lattice induces geometrical frustration in both systems, which leads to the critical behavior rather different from their ferromagnetic counterparts. We confirm second-order phase transitions belonging to the standard Ising universality class occurring at higher temperatures, however, in both models these change at tricritical poin…
We consider the ferromagnetic Ising model on the Cayley tree and we investigate the decomposition of the free state into extremal states below the spin glass temperature. We show that this decomposition has uncountably many components. The tail observable showing that the free state is not extremal is related to the Edwards-Anderson parameter, measuring the variance of the (random) magnetization obtained from drawing boundary conditions from the free state.
The Ising model is an equilibrium stochastic process used as a model in several branches of science including magnetic materials, geophysics, neuroscience, sociology and finance. Real systems of interest have finite size and a fixed coupling matrix exhibiting quenched disorder. Exact methods for the Ising model, however, employ infinite size limits, translational symmetries of lattices and the Cayley tree, or annealed structures as ensembles of networks. Here we show how the Ising partition function can be evaluated exactly by exploiting small tree-width. This structural property is exhibited …
The Monte Carlo analysis for the magnetic response of a single-walled nanotube using the Metropolis and Wang Landau algorithms is reported in the present paper. The nanotube architecture used in the present study utilizes the spin half Ising model with nearest neighbors interaction and obtained various magnetic orderings namely, ferromagnetic, G-type anti-ferromagnetic, A-type anti-ferromagnetic, and C-type anti-ferromagnetic. It is also found that the phase changes from ferromagnetic/anti-ferromagnetic to paramagnetic with the modification of system's control parameters. The transition temper…
Critical temperatures for the ferro-paramagnetic transition in the Ising model are evaluated for five Archimedean lattices, basing on Monte Carlo simulations. The obtained Curie temperatures are 1.25, 1.40, 1.45, 2.15 and 2.80 [J/k_B] for (3,12^2), (4,6,12), (4,8^2), (3,4,6,4) and (3^4,6) lattices, respectively.
We present exact results for the critical behavior of the RFIM on complete graphs and trees, both at equilibrium and away from equilibrium, i.e., models for hysteresis and Barkhausen noise. We show that for stretched exponential and power law distributions of random fields the behavior on complete graphs is non-universal, while the behavior on Cayley trees is universal even in the limit of large co-ordination.
In this work we try to use the Monte Carlo algorithm, metropolis, to study the behavior of 2D magnetic systems; honeycomb, hexagonal and square lattices. In this study we use Ising Model of magnetism, with considering only nearest neighbors in the Hamiltonian and with the Ferromagnetic correlation constant. The code was written in Payton. In this calculation, by comparing the results obtained for all lattices, we will show that the paramagnetic to ferromagnetic phase transition depends on the number of neighbor and it is a direct dependence.
The specific heat of the two-dimensional $\pm J$ Ising model has been investigated by the numerical transfer matrix method and Monte Carlo simulations from a new point of view. The region where a part of the specific heat takes the negative value has been investigated, which is characteristic of frustrated systems and reflects the non-trivial degeneracy of the ground state. The region mentioned above is found to be fairly large in the $p-T$ plane ($p$ is the concentration of the ferromagnetic bond and $T$ is the temperature). Moreover, it includes the Nishimori line. Namely, it includes a part…
We demonstrate the feasibility of ground state preparation for the transverse Ising model using projective cooling, and show that the algorithm can effectively construct the ground state in the disordered (paramagnetic) phase. On the other hand, significant temperature effects are encountered in the ordered (ferromagnetic) phase requiring larger lattices to accurately simulate.
Emergence of a Boundary-Sensitive Phase in Hyperbolic Ising Models
Boston College（波士顿学院）；Washington University（华盛顿大学）；University of Oxford（牛津大学）；Technische Universität Chemnitz（开姆尼茨工业大学）；Indian Institute of Technology Madras（印度马德拉斯理工学院）；Institute for Advanced Study, Princeton（普林斯顿高等研究院）；Indiana University（印第安纳大学）
The continuous ferromagnetic-paramagnetic phase transition in the two-dimensional Ising model has already been excessively studied by conventional canonical statistical analysis in the past. We use the recently developed generalized microcanonical inflection-point analysis method to investigate the least-sensitive inflection points of the microcanonical entropy and its derivatives to identify transition signals. Surprisingly, this method reveals that there are potentially two additional transitions for the Ising system besides the critical transition.
The Ising model in uncorrelated scale-free networks has been studied by means of Monte Carlo simulations. These networks are characterized by a degree (or connectivity) distribution $P(k) \sim k^{-γ}$. The ferromagnetic-paramagnetic transition temperature has been studied as a function of the parameter $γ$. For $γ> 3$ our results agree with earlier analytical calculations, which found a phase transition at a temperature $T_c(γ)$ in the thermodynamic limit. For $γ\leq 3$, a ferromagnetic-paramagnetic crossover occurs at a size-dependent temperature $T_{co}$, and the system is in the ordered fer…
The mean field solution of the Ising model on a Barabasi-Albert scale-free network with ferromagnetic coupling between linked spins is presented. The critical temperature $T_c$ for the ferromagnetic to paramagnetic phase transition (Curie temperature) is infinite and the effective critical temperature for a finite size system increases as the logarithm of the system size in agreement with recent numerical results of Aleksiejuk, Holyst and Stauffer.
We derive an exact closed-form expression for fidelity susceptibility of the quantum Ising model in the transverse field. We also establish an exact one-to-one correspondence between fidelity susceptibility in the ferromagnetic and paramagnetic phases of this model. Elegant summation formulas are obtained as a by-product of these studies.
We consider the procedure for calculating the pair correlation function in the context of the Cluster Variation Methods. As specific cases, we study the pair correlation function in the paramagnetic phase of the Ising model with nearest neighbors, next to the nearest neighbors and plaquette interactions in two and three dimensions. In presence of competing interactions, the so called disorder line separates in the paramagnetic phase a region where the correlation function has the usual exponential behavior from a region where the correlation has an oscillating exponentially damped behavior. In…
We show that, above the critical temperature, if the dimension D of a given Ising spin glass model is sufficiently high, its free energy can be effectively expressed through the free energy of a related Ising model. When, in a large sense, D is infinite, in the paramagnetic phase and on its boundary the mapping is exact. In this limit the method provides a general and simple rule to obtain exactly the upper phase boundaries. We provide even simple effective rules to find crossover surfaces and correlation functions. We apply the mapping to several spin glass models.
We investigate phase transitions in the Ising model and the ANNNI model in transverse field using the interface approach. The exact result of the Ising chain in a transverse field is reproduced. We find that apart from the interfacial energy, there are two other response functions which show simple scaling behaviour. For the ANNNI model in a transverse field, the phase diagram can be fully studied in the region where a ferromagnetic to paramagnetic phase transition occurs. The other region can be studied partially; the boundary where the antiphase vanishes can be estimated.
We apply a recently developed theory for metastability in open quantum systems to a one-dimensional dissipative quantum Ising model. Earlier results suggest this model features either a non-equilibrium phase transition or a smooth but sharp crossover, where the stationary state changes from paramagnetic to ferromagnetic, accompanied by strongly intermittent emission dynamics characteristic of first-order coexistence between dynamical phases. We show that for a range of parameters close to this transition/crossover point the dynamics of the finite system displays pronounced metastability, i.e.,…
The critical properties of the one-dimensional transverse Ising model in the presence of a longitudinal magnetic field were studied by the quantum fidelity method. We used exact diagonalization to obtain the ground-state energies and corresponding eigenvectors for lattice sizes up to 24 spins. The maximum of the fidelity susceptibility is used to locate the various phase boundaries present in the system. The type of dominant spin ordering for each phase was identified by examining the corresponding ground-state eigenvector. For a given antiferromagnetic nearest-neighbor interaction J2, we calc…
Phase diagrams and spontaneous magnetization are rigorously calculated for a coupled spin-electron model on a doubly decorated honeycomb lattice, which accounts for a quantum-mechanical hopping of the mobile electrons on decorating sites, the nearest-neighbor Ising coupling between mobile electrons and localized spins, as well as, the further-neighbor Ising coupling between the localized spins placed on nodal sites. The spontaneously ordered ferromagnetic phase, spontaneously ordered antiferromagnetic phase and disordered paramagnetic phase emerge in a phase diagram depending on an electron fi…
The Grüneisen ratio (GR) has emerged as a superb tool for the diagnosis of quantum phase transitions, which diverges algebraically upon approaching critical points of continuous phase transitions. However, this paradigm has been challenged recently by observations of a finite GR for self-dual criticality and divergent GR at symmetry-enhanced first-order transitions. To unveil the fascinating GR further, we exemplify the idea by studying an exactly solvable quantum Ising model with Dzyaloshinskii-Moriya interaction, which harbors a ferromagnetic phase, a paramagnetic phase, and a chiral Lutting…
We study a 3-dimensional Ising model in which the tendency to order due to short-range ferromagnetic interactions is frustrated by competing long-range (Coulombic) interactions. Complete ferromagnetic ordering is impossible for any nonzero value of the frustration parameter, but the system displays a variety of phases characterized by periodically modulated structures. We have performed extensive Monte-Carlo simulations which provide strong evidence that the microphase separation transition between paramagnetic and modulated phases is a fluctuation-induced first-order transition. Additional tr…
The two-dimensional (2D) random-bond Ising model has a novel multicritical point on the ferromagnetic to paramagnetic phase boundary. This random phase transition is one of the simplest examples of a 2D critical point occurring at both finite temperatures and disorder strength. We study the associated critical properties, by mapping the random 2D Ising model onto a network model. The model closely resembles network models of quantum Hall plateau transitions, but has different symmetries. Numerical transfer matrix calculations enable us to obtain estimates for the critical exponents at the rand…
A frustrated Ising model on a diamond hierarchical lattice is studied. We obtain the exact partition function of this model and calculate the transition temperature, specific heat, entropy, magnetization, and ferromagnetic correlation function. Depending on the magnitude of a parameter giving the frustration, there exist three types of ground states: ferromagnetic, classical spin-liquid with highly developed short-range order, and paramagnetic. The dependence of the zero-temperature entropy on the frustration parameter has an infinite number of steps. The temperature dependence of the specific…
We apply the Bethe-Peierls approximation to the problem of the inverse Ising model and show how the linear response relation leads to a simple method to reconstruct couplings and fields of the Ising model. This reconstruction is exact on tree graphs, yet its computational expense is comparable to other mean-field methods. We compare the performance of this method to the independent-pair, naive mean- field, Thouless-Anderson-Palmer approximations, the Sessak-Monasson expansion, and susceptibility propagation in the Cayley tree, SK-model and random graph with fixed connectivity. At low temperatu…
We study the effects of dissipation on a randomly dilute transverse-field Ising magnet at and close to the percolation threshold. For weak transverse fields, a novel percolation quantum phase transition separates a super-paramagnetic cluster phase from an inhomogeneously ordered ferromagnetic phase. The properties of this transition are dominated by large frozen and slowly fluctuating percolation clusters. Implementing numerically a strong-disorder real space renormalization group technique, we compute the low-energy density of states which is found to be in good agreement with the analytical …
In an artificial 3D percolation nano medium, the clusters filled by the Ising magnets give rise to a topologically nontrivial magnetic structure, leading to new features of the ferromagnetic phase transition without an external magnetic field. In such an inhomogeneous system, the standard Ising model is strongly modified by the spatial percolation cluster distribution. We found numerically that at percolation occupation probability $p<1$ far from the percolation threshold $p_{c3D}$, the magnetization shows ferromagnetic-paramagnetic phase transition with the transition temperature $T_{c}$ depe…
A two-dimensional fluid of hard spheres each having a spin $\pm 1$ and interacting via short-range Ising-like interaction is studied near the second order phase transition from the paramagnetic gas to the ferromagnetic gas phase. Monte Carlo simulation technique and the multiple histogram data analysis were used. By measuring the finite-size behaviour of several different thermodynamic quantities,we were able to locate the transition and estimate values of various static critical exponents. The values of exponents $β/ν$ and $γ/ν$ are close to the ones for the two-dimensional lattice Ising mode…
