Hamiltonian Ising spin

In a computational study, the values of Ja and J e can be obtained by mapping the total energies of different long-range ordered magnetic arrangements onto an Ising spin Hamiltonian of the form... 

Just as the magnetic state energies can be mapped onto an Ising spin Hamiltonian, so may the energies of various crystal field split states be mapped onto the Kanamori Hamiltonian [89]. For an isolated metal site in which bonding and magnetic interactions are neglected, this takes the form... 

There is a fundamental relationship between d-dimensional PCA and d + 1)-dimensional Ising spin models. The simplest way to make the connection is to think of the successive temporal layers of the PCA as successive hyper-planes of the next higher-dimensional spatial lattice. Because the PCA rules (at least the set of PCA rules that we will be dealing with) are (1) Markovian (i.e. the probability of a state at time t + T depends only on a set of states at time t, and (2) local, one can always define a Hamiltonian on the higher-dimensioned spatial lattice such that the thermodynamic weight of a configuration 5j,( is equal to the probability of a corresponding space-time history Si t). ... 

It was shown by Wilson [131] that the Kadanoff procedure, combined with the Landau model, may be used to identify the critical point, verify the scaling law and determine the critical exponents without obtaining an exact solution, or specifying the nature of fluctuations near the critical point. The Hamiltonian for a set of Ising spins is written in suitable units, as before... 

The Hamiltonian of an Ising spin glass, given by Edwards and Anderson (EA) [78], is... 

Before presenting the measurements, we summarize a simplified spin Hamiltonian describing the [Mn4]2 dimer [48], Each Mn4 SMM can be modeled as a giant spin of S -9/2 with Ising-like anisotropy [Eq. (1)]. The corresponding Hamiltonian is given by... 

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-... 

As is well known, the lattice gas model can be rewritten in terms of an equivalent Ising Hamiltonian W/sing by the transformation c, = (1 — )/2, which maps the two choices c, = 0,1 to Ising spin orientations St — 1. In our example this yields (Binder and Landau, 1981)... 

Some compounds exhibit a spin crossover of the form that the fraction xHs of molecules in the HS state increases with temperature in two steps a plateau of a few K exists between these steps. This behaviour can be explained by considering two sublattices (A and B) containing the same number of molecules [26], The Ising-like Hamiltonians corresponding to the respective lattices, in the mean field approach, are defined as follows... 

An important aspect of one-dimensional magnetism is the nature of the magnetic interaction between the spins. It can be isotropic, as described by the spin Hamiltonian Equation (6), or anisotropic. In the axial limit the anisotropy can be of the easy-axis type (Ising anisotropy) or of the easy-plane type (XY anisotropy). The former means that the spins tend to orient parallel to a preferential direction while for the latter they orient in a plane. The physical properties of the chains are strongly influenced by the nature of the anisotropy. ... 

The Hamiltonian of an Ising spin system in an external field B is expressed as... 

The Hamiltonian of the CE is a spin Hamiltonian on a lattice and a generaHzation of the simple Ising model. In the simplest case, the only difference to Equation 11.9... 

Fig. 7.5 Coupling constants between the spins of an elementary plaquette of the triangular lattice used in defining the Ising Hamiltonian appearing in equation 7.65.

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. ... 

The lattice dimensionality of the interacting spin system describes the number of directions in which there are extended interaction pathways. Along with the spin dimensionality, the lattice dimensionality determines if a magnetic system can spontaneously undergo a transition to a magnetically ordered state at a finite temperature (see Section 2.6). Spin and lattice dimensionalities are independent parameters so within each class of lattice dimensionality, the spins can be described by Ising-like, XY-like, or Heisenberg Hamiltonians. 

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,... 

---