Hamiltonian Ising

The elimination of the anisotropic part in Eq. 3.22 leads to the Heisenberg Hamiltonian for isotropic magnetic interactions. The spins are considered as co-linear vectors whose principal quantization axis has no spatially preferred orientation. An even simpler model Hamiltonian can be obtained by putting Axx and Ayy to zero in Eq. 3.21. Then, the spin reduces to a classical vector whose orientation in space is not defined and the resulting model Hamiltonian describes the isotropic coupling of two (anti-)parallel spins. Replacing A z by -J, the following expression is obtained 

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Ab-initio studies of surface segregation in alloys are based on the Ising-type Hamiltonian, whose parameters are the effective cluster interactions (ECI). The ECIs for alloy surfaces can be determined by various methods, e.g., by the Connolly-Williams inversion scheme , or by the generalized perturbation method (GPM) . The GPM relies on the force theorem , according to which only the band term is mapped onto the Ising Hamiltonian in the bulk case. The case of macroscopically inhomogeneous systems, like disordered surfaces is more complex. The ECIs can be determined on two levels of sophistication ... 

The parameters of the semi-infinite alloy Ising Hamiltonian are the configurationally independent part of the alloy internal energy Eq, the on-site energies the interatomic pair interactions and generally, interatomic interactions of higher order. 

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.

Applying exactly the same reasoning to our stochastic net, but using equation 10.9 for the Hopfield energy in place of the Ising Hamiltonian, we obtain the analogous expression... 

The theoretical foundation for describing critical phenomena in confined systems is the finite-size scaling approach [64], by which the dependence of physical quantities on system size is investigated. On the basis of the Ising Hamiltonian and finite-size scaling theory, Fisher and Nakanishi computed the critical temperature of a fluid confined between parallel plates of distance D [66]. The critical temperature refers to, e.g., a liquid/vapor phase transition. Alternatively, the demixing phase transition of an initially miscible Kquid/Kquid mixture could be considered. Fisher and Nakashini foimd that compared with free space, the critical temperature is shifted by an amoimt... 

Note that for = 2 both Eqs. (17), (18) essentially reduce again to the Ising Hamiltonian, Eq. (9), with nearest neighbor interaction only. The latter model is described by the following critical behavior for its order parameter if/, ordering susceptibility and specific heat C ... 

The average <(poz)3>=, or order parameter, is calculated over the equilibrium density matrix, p=e H/T/Tr[e H/T], where Tr denotes the trace (sum over diagonal elements). The Ising Hamiltonian can be expressed as ... 

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. ... 

Thermodynamic properties of non-regular chains have been described in reference [27]. In that case, several correlation functions T) should be introduced. Taking the simple example of a chain of spin S with an alternation of 7i and J2 magnetic exchanges (i.e. a chain of dimers with Ji and J2 being the intra-dimer and inter-dimer interactions, respectively), the corresponding Ising Hamiltonian reads ... 

Another important class of effective Hamiltonians saw action in the context of phase diagrams in chap. 6. In particular, we noted that with the complexity added by chemical disorder the only way to effect a systematic search over entire classes of structural competitors was to invoke an extended Ising Hamiltonian of the form... 

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)... 

Here we have omitted constant terms in the free energy and anticipated that a molecular field treatment of an Ising Hamiltonian that describes exactly the situation considered here, namely eq. (1), is consistent with eq. (220). The coefficient of the linear term in eq. (220) thus describes the action of a local field (i.e., the variable conjugate to the local order parameter) right at the hard walls (or free surfaces, respectively). If

order parameter of gas-liquid condensation, the field H can be interpreted as the binding potential of particles at the hard wall. 

As a prelude to discussing mean-field theory, we review the solution for non-interacting magnets by setting J = 0 in the Ising Hamiltonian. The PF... 

Let us consider a binary system A — B. Its particular configuration is determined by a set of occupation indices i/i, where /, — 1 if the site i is occupied by an atom of the type A, and / = 0 otherwise. This form corresponds to the lattice gas model. The effective concentration-independent Ising Hamiltonian reads... 

Table 2. Difference of the on-site terms A = D — D (in mRy) of the effective Ising Hamiltonian for a (001) surface of fee Ag25Pd7s as calculated within model II for the converged inhomogeneous concentration profile. Individual contributions from the band energy, core electrons, double-counting terms, Madelung energy and non-spherical corrections, both electrostatic (nse) and exchange-correlation (nsxc) aie given. Their sum, is compared with the lesult of model I, A. ...

The observable of interest, in this case, is the relative spin orientation on neighbouring Mn ions, which will be investigated using a model Ising Hamiltonian, as described earlier. 

To calculate the correlation functions for composition fluctuations in a binary system, consider the Ising Hamiltonian of Eq. (2.17) with an additional term A,-5,- where A,- is a local field. Show that the correlation function... 

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