Ising model Hamiltonian

Onsager s solution to the 2D Ising model in zero field (H= 0) is one of the most celebrated results in theoretical chemistry [105] it is the first example of critical exponents. Also, the solution for the Ising model can be mapped onto the lattice gas, binary alloy and a host of other systems that have Hamiltonians that are isomorphic to the Ising model Hamiltonian. 

The FCEM expressions for binary alloys were obtained using the Ising model Hamiltonian and an expansion of the partition function and free energy in terms of solute concentration [1,3,19]. The free energy of a binary alloy (A -solute, S - solvent) reads. 

For 7=1 the Hamiltonian reduces to the Ising model and for y = 0 to the XY model. For the pure homogeneous case, /,, +i = / and = B, the system exhibits a quantum phase transition at a dimensionless coupling constant... 

The Ising-type Hamiltonian is usually employed for H-bonded ferroelectrics. In the more general quantum-mechanical approach (dynamic Ising model [2,3]) it has a form ... 

If b = 0, a = 1 the Ising model, if b = 1, a = 1 the isotropic Heisenberg model, and if b = 1, a = 0, the planar Heisenberg model or X-Y model is valid. In the following, the values of the exchange constants were calculated, or recalculated assuming a spin Hamiltonian of this form, instead of = - J 2 SjSi, occasionally used by certain au-... 

The Ising model assumes the magnetic interactions to be anisotropic. In fact, this phenomenon does not occur practically and the choice of an Ising Hamiltonian will be made on the basis of other factors such as the presence of a crystal field or a magnetic dipolar field both of which can polarize the spin in a certain direction of the crystal. The first case very often results from the existence of an orbital momentum7. ... 

Therefore, the consistent study of many-particle system dynamics should start by establishing the Hamiltonian H and then solving the evolution equation (93). Unfortunately, examples of such calculations are very rare and are only valid for limited classes of model systems (such as the Ising model) since these... 

We compared the linearized form Eqn. (IV.6) with the long-range Ising-model-like fitness function (cf. the Hamiltonian in Section III.5)... 

In order to describe the collective-update schemes that are the focus of this chapter, it is necessary to introduce the Ising model. This model is defined on a d-dimensional lattice of linear size L (a square lattice in d = 2 and a cubic lattice in d = 3) with, on each vertex of the lattice, a one-component spin of fixed magnitude that can point up or down. This system is described by the Hamiltonian,... 

Although the Heisenberg and Ising models have been intensively used in theoretical works, the description of real materials often requires more complicated Hamiltonians. For example when single-ion anisotropy is relevant, a finite magnetic anisotropy has to be considered. In this case, the corresponding Hamiltonian can be written ... 

Statistical Mechanics of the Ising Model. A tremendously important model within statistical mechanics at large, and for materials in particular, is the Ising model. This model is built around the notion that the various sites on a lattice admit of two distinct states. The disposition of a given site is represented through a spin variable a which can take the values 1, with the two values corresponding to the two states. With these kinematic preliminaries settled, it is possible to write the energy (i.e. the Hamiltonian) as a function of the occupations of these spin variables as... 

Partition Function for the One-Dimensional Ising Model For the one-dimensional Ising model with near-neighbor interactions with a Hamiltonian of the form (i.e. zero field)... 

The emergence of the types of polytypes described above may be rationalized on the basis of a variety of different ideas. Rather than exploring the phenomenon of polytypism in detail, we give a schematic indication of one approach that has been used and which is nearly identical in spirit to the effective Hamiltonian already introduced in the context of stacking fault energies (see eqn (9.18)). In particular, we make reference to the so-called ANNNI (axial next-nearest-neighbor Ising) model which has a Hamiltonian of the form... 

The latter situation in fact is predicted for the so-called ANNNI (axial next-nearest neighbor Ising) model (Selke, 1988,1992). This Ising model has a competing interaction Jj < 0 in one lattice direction only, and thus the Hamiltonian is... 

When the Ising chain (SY spins) interacts substantially with another one (5, spins), forming thus a ladder-type structure, the model Hamiltonian is extended for an interchain coupling parameter. In the simplest case of the S) = Si = 1/2 system the representation of the transfer matrix has the dimension k = 4 x 4. Its eigenvalues are then obtained only numerically and thus the magnetisation and the magnetic susceptibility cannot be represented by an analytic function. However, numerical solution is accessible using computers [17]. 

Many important problems in computational physics and chemistry can be reduced to the computation of dominant eigenvalues of matrices of high or infinite order. We shall focus on just a few of the numerous examples of such matrices, namely, quantum mechanical Hamiltonians, Markov matrices, and transfer matrices. Quantum Hamiltonians, unlike the other two, probably can do without introduction. Markov matrices are used both in equilibrium and nonequilibrium statistical mechanics to describe dynamical phenomena. Transfer matrices were introduced by Kramers and Wannier in 1941 to study the two-dimensional Ising model [1], and ever since, important work on lattice models in classical statistical mechanics has been done with transfer matrices, producing both exact and numerical results [2]. 

Niedermayer pointed out that it is not necessary to constrain the links with which we make clusters to be only between spins which are pointing in the same direction. In general, we can define two different probabilities for putting links between sites—one for parallel spins and one for antiparallel ones. The way Niedermayer expressed it, he considered the energy contribution Ey that a pair of spins i and j makes to the Hamiltonian. In the case of the Ising model, for example... 

---