Ising model on the generalized Bruhat-Tits tree

The partition function and the correlation functions of the Ising model on the generalized Bruhat-Tits tree are calculated. We computed also the averages of these correlation functions when the corresponding vertices are attached to the boundary of the generalized Bruhat-Tits tree.     

Theory of the Ising model on a Cayley tree

The Ising model on a Cayley tree is known to exhibit a phase transition of continuous order. In this paper we present a complete and quantitative analysis of the leading singular term in the free energy which is associated with this phase transition. We have been able to solve this problem by considering the distribution of zeros of the partition function. The most interesting new feature in our results is a contribution to the free energy which performs singular oscillations as the magnetic field approaches zero.     

Magnetization of the Ising model on the generalized checkerboard lattice

We consider the Ising model on the generalized checkerboard lattice. Using a recent result by Baxter and Choy, we derive exact expressions for the magnetization of nodal spins at two values of the magnetic field,H=0 andH=i1/2kT. Our results are given in terms of Boltzmann weights of a unit cell of the checkerboard lattice without specifying its cell structures.     

Yang-Lee zeros of the Ising model on the closed Cayley tree

The Yang-Lee zeros of the partition function of the ferro-, antiferro- and of the partially antiferromagnetic anisotropic Ising models defined on the closed symmetric Cayley tree are studied. The applicability of the Yang-Lee theorem to the antiferromagnetic systems is shown to be a consequence of the invariance of the unit circle under the Bethe-Peierls map. The relationship as well as the distinction between the set of zeros and the Julia set is established. The fractal dimension of the Julia set is shown to be equal to one in the low temperature phase and to be a decreasing function of the temperature in the paramagnetic phase of the three systems.     

Three-spin correlation of the Ising model on the generalized checkerboard lattice

The Ising model on the generalized checkerboard lattice is studied and the three-spin correlation function is obtained for the three nodal spins surrounding a unit cell of the checkerboard lattice. As an application of this result, the spontaneous magnetization of the internal spin within a unit cell is calculated.     

Ising model with competing interactions on a Cayley tree

An iterative scheme is developed for a renormalized effective nearest-neighbor couplingK _r and effective field per siteK _r for spins in therth shell of a Cayley tree with nearest neighborJ, and next nearest neighborJ′, interactions between Ising spins on the lattice. In addition to the expected paramagnetic, ferromagnetic, and antiferromagnetic phases, we find an intermediate range ofJ'/J < 0 values whereX _r, and K_r iterate to a continuous or quasicontinuous attractor in theX-K plane. In this range the local magnetization is mainly chaotic with oscillatory glasslike behavior. Embedded in the chaos, however, are regions of periodic and commensurate phases.     

Susceptibility divergence of the spin 1 Ising model on the Cayley tree

The temperature T₂(1), below which the zero-field isothermal susceptibility diverges for the spin 1 Ising model on the M-generation Cayley tree, in the limit as M → ∞, is located. The general approach follows closely that of Falk for the spin $\text{1}\text{2}$ problem. Pruning and retraction identities are used to find upper and lower bounds on the bond-length dependence of the general two-spin correlation function and thence on the susceptibility. A conjecture is advanced regarding the divergence of the higher-order susceptibilities.     

Magnetic hysteresis in the generalized mean-field model for Ising system

Magnetic hysteresis and demagnetization of a simple cubic lattice of Ising spins is studied in the framework of the generalized mean-field approximation taking into account spatial fluctuations of the local magnetic field. The existence of a dynamic phase transition is demonstrated.     

Divergence of the higher-order susceptibilities of the random-site Ising model on the cayley tree

For the random-site Ising model on the Cayley tree, the highest temperatures are given at which expressions for the higher order susceptibilities, the higher derivatives of the free energy with respect to the external field, diverge.     

Phase transitions of continuous order: Ising model on a Cayley tree

We introduce the notion of a phase transition of continuous order. As a model which displays such a phase transition an Ising model on a Cayley tree is solved exactly and discussed in detail.     

The Ising model on a closed Cayley tree: Zeros of the partition function

It is shown that the zeros of the partition function of an Ising model defined on the closed Cayley tree have a particularly simple structure in the complex interface interaction plane.     

Spin-glass and spin-crystal phase transition of a regular Ising model on the Cayley tree

A regular Ising model with nearest-neighbour interaction of +J and -J on a Cayley tree is reported to show a phase transition from the paramagnetic phase to the spin-glass and spin-crystal phase in a uniform external field below a critical value h_c.     

Higher-order susceptibilities of the regular and the random Ising model on the Cayley tree. I

Explicit expressions for the fourth-order susceptibility (⁴), the fourth derivative of thebulk free energy with respect to the external field, are given for the regular and the random-bond Ising model on the Cayley tree in the thermodynamic limit, at zero external field. The fourth-order susceptibility for the regular system diverges at temperature T _c ⁽⁴⁾ = 2k _B ^–1 J/ln{1+2/[(z–1)^3/4–1]}, confirming a result obtained by Müller-Hartmann and Zittartz [Phys. Rev. Lett. 33:893 (1974)]; Herez is the coordination number of the lattice,J is the exchange integral, andk _B is the Boltzmann constant. The temperatures at which ⁽⁴⁾ and the ordinary susceptibility (²) diverge are given also for the random-bond and the random-site Ising model and for diluted Ising models.     

Third and fourth order phase transitions: Exact solution for the Ising model on the Cayley tree

In this work we present the first exact solution of a system of interacting particles with phase transitions of order higher than two. The presented analytical derivation shows that the Ising model on the Cayley tree exhibits a line of third order phase transition points, between temperatures and , and a line of fourth order phase transitions between T_BP and ∞, where k_B is the Boltzmann constant, and J is the nearest-neighbor interaction parameter.     

New perspectives on the Ising model

The Ising model, in presence of an external magnetic field, is isomorphic to a model of localized interacting particles satisfying the Fermi statistics. By using this isomorphism, we construct a general solution of the Ising model which holds for any dimensionality of the system. The Hamiltonian of the model is solved in terms of a complete finite set of eigenoperators and eigenvalues. The Green’s function and the correlation functions of the fermionic model are exactly known and are expressed in terms of a finite small number of parameters that have to be self-consistently determined. By using the equation of the motion method, we derive a set of equations which connect different spin correlation functions. The scheme that emerges is that it is possible to describe the Ising model from a unified point of view where all the properties are connected to a small number of local parameters, and where the critical behavior is controlled by the energy scales fixed by the eigenvalues of the Hamiltonian. By using algebra and symmetry considerations, we calculate the self-consistent parameters for the one-dimensional case. All the properties of the system are calculated and obviously agree with the exact results reported in the literature.     

Effective-field theory on the kinetic Ising model

As an analytical method, the effective-field theory (EFT) is used to study the dynamical response of the kinetic Ising model in the presence of a sinusoidal oscillating field. The effective-field equations of motion of the average magnetization are given for the square lattice (Z=4) and the simple cubic lattice (Z=6), respectively. The dynamic order parameter, the hysteresis loop area and the dynamic correlation are calculated. In the field amplitude h₀/ZJ-temperature T/ZJ plane, the phase boundary separating the dynamic ordered and the disordered phase has been drawn, and the dynamical tricritical point has been observed. We also make the compare results of EFT with that given by using the mean field theory (MFT).     

One-dimensional Ising chain with generalized molecular field

The behavior of a spin-$\text{1}\text{2}$ one-dimensional nearest-neighbor Ising chain coupled to a generalized internal molecular field is considered. The internal field is shown to be equivalent to long-range spin-spin interactions. An assumed expansion of the internal field in powers of the magnetization results in the possibility of first- or second-order phase transitions and softening in the spontaneous magnetization and specific heat near the transition temperature. The internal energy and magnetic susceptibility are also discussed.