Hamiltonian Ising-like

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... 

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. 

When the exchange Hamiltonian is no longer isotropic, that is, when a /3 in the exchange Hamiltonian, the susceptibility is no longer isotropic. The anisotropic Hamiltonian describes a system in which there is a preferred direction (Ising-like, a < P) or preferred plane (XY-like, a > P) irv space. The magnetic response will be different depending on whether the external field is applied parallel to or perpendicular to the preferred direction(s). These responses correspond to the parallel (xy) and perpendicular susceptibilities (Xi). 

Fig. 9.37. Ranking of structural energies associated with different polytypes using the Ising-like effective Hamiltonian described in the text (adapted from Engel and Needs (1990)).

Here fictitious spin with the eigenvalues a = -1 for the LS state and a = +1 for the HS state. The postulated Hamiltonian formally describes an Ising-like system under the external field At. Unlike true magnetic systems, the degeneracies of the two levels are different thus the suffix Ising-like. Within the mean field approach the single molecule can be characterised by the Hamiltonian... 

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... 

Within the Ising-like model the Hamiltonian of a dinuclear system formed of a symmetric pair of metallic centres is... 

Similar to the approach of cluster expansion for the description of three-dimensional structures [77] (see also Chapter 11, in Volume 3), the total energy of a given distribution can be expressed via an Ising-like 2D lattice gas Hamiltonian of the form [77, 82]... 

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 ... 

In the opposite end, in the strongly correlated domain, magnetic behavior was studied via Ising and Heisenberg-related empirical Hamiltonians, with actual results mainly semi-classical or mean-field-like. 

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

Like SAWs that are related to the 0(m)-model, percolation is connected to the q-state Potts model. The g-state Potts model [86] is a spin model whose high temperature expansion (when g 1) corresponds to diagr lms that are percolation configurations. In the Potts model to the each site a of a regulcu lattice a spin variable Ox corresponds which can be in q possible states (g = 2 corresponds to the Ising case). Interactions favor nearest neighbors which are in the same state. The effective Hamiltonian for the model reads ... 

Many particles systems are usually described by a class of Hamiltonians that are composed of local interactions, i.e. between a small number of particles, like the Ising and Heisenberg models ... 

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