Effective spin model

When gi = g2 and a>i = C02, this model is reduced to the iJ (g) e IT system. For simplicity, we assume that g /a>i 2/ 2 and treat the g2 term within second-order perturbation. By adopting a similar method in treating the Holstein model, we can map he E (bi + >2) model into the effective spin model as... 

Fig. 14 Schematic representation of the effective spin model for the <8> (hi + 62) molecular crystal system in one dimension. y, y = a, P p = x, y, z) denotes the interaction between nearest-neighbor spins y and y. is the on-site interaction between a and p at the same site...

In one dimension, this spin model is represented by a two-leg ladder system [120] as shown in Fig. 14 and examples of possible phases are schematically given in Fig. 15. In two dimensions, we may think of the effective spin model as shown in Fig. 16. As we see, those spin models are the subject of intense researches in relation to HTSC and at present we cannot give a further reliable information. 

Fig. 16 Schematic representation of the effective spin model for the E (b 2) molecular...

An even coarser description is attempted in Ginzburg-Landau-type models. These continuum models describe the system configuration in temis of one or several, continuous order parameter fields. These fields are thought to describe the spatial variation of the composition. Similar to spin models, the amphiphilic properties are incorporated into the Flamiltonian by construction. The Flamiltonians are motivated by fiindamental synnnetry and stability criteria and offer a unified view on the general features of self-assembly. The universal, generic behaviour—tlie possible morphologies and effects of fluctuations, for instance—rather than the description of a specific material is the subject of these models. 

This example shows that dipolar interactions can produce unexpected effects in systems containing polynuclear clusters, so that their complete quantitative description requires a model in which the dipolar interactions between all the paramagnetic sites of the system are explicitly taken into account. Local spin models of this kind can provide a description of the relative arrangement of the interacting centers at atomic resolution and have been worked out for systems containing [2Fe-2S] and [4Fe-4S] clusters (112, 192). In the latter case, an additional complication arises due to the delocalized character of the [Fe(III), Fe(II)] mixed-valence pair, so that the magnetic moments carried by the two sites A and B of Fig. 8B must be written... 

Then, there are model Hamiltonians. Effectively a model Hamiltonian includes only some effects, in order to focus on those effects. It is generally simpler than the true full Coulomb Hamiltonian, but is made that way to focus on a particular aspect, be it magnetization, Coulomb interaction, diffusion, phase transitions, etc. A good example is the set of model Hamiltonians used to describe the IETS experiment and (more generally) vibronic and vibrational effects in transport junctions. Special models are also used to deal with chirality in molecular transport junctions [42, 43], as well as optical excitation, Raman excitation [44], spin dynamics, and other aspects that go well beyond the simple transport phenomena associated with these systems. 

In Kadanoff s [130, 131] two-dimensional block-spin model four neighbouring spins are assumed to have identical spins, either up or down, near the critical point. The block of four then acts like a single effective spin. The lattice constant of the effective new lattice is double the original lattice constant. The coherence length measured in units of the new lattice constant will hence be at half of its original measure. Repetition of this procedure allows further reduction in by factors of two, until finally one has an effective theory with = 1. At each step it is convenient to define renormalized block spins such that their magnitude is 1 instead of 4. The energy of such blocked spins is... 

D. Maynau, P. Durand, J. P. Duadey, and J. P. Malrieu, Phys. Rep. A, 28, 3193 (1983). Direct Determination of Effective-Hamiltonians by Wave-Operator Methods. 2. Application to Effective-Spin Interactions in -Electron Systems. P. Durand and J. P. Malrieu, in Advances in Chemical Physics (Ah Initio Methods in Quantum Chemistry—I), K. P. Lawley, Ed., Wiley, New York, 1987, Vol. 67, pp. 321-412. Effective Hamiltonians and Pseudo-Operators as Tools for Rigorous Modelling. 

The REM, NK and p-spin models all are attempts to capture the important statistical properties of true molecular landscapes in a simple model. Because they contain no biophysical information, they are limited in how well they can achieve this. The block model is an important step in removing some of the simplifications in these models, as it allows for nonstationary properties that can be matched to different regions of molecules. Ideally, landscape models can be based on experimental data. Unfortunately, despite the tremendous interest in molecular optimization, there is still relatively little data that can be used this way. As more data are collected on the effects of substitutions in protein structural and loop regions, antibody CDRs and framework regions, etc., a block or other type of model can be developed that uses appropriate fitness functions for each block. Combined efforts by theoreticians and experimentalists may also help devise experiments that measure key true affinity landscape properties without excessive laboratory effort. 

This model assumes that the Mn ions occupy the sites of a cubic lattice (taken to be of unit lattice parameters), while the dopant A ions occupy an x fraction of the body centre sites of each unit cube formed by the Mn ions. Since the aim is to study the effect of long-range Coulomb interactions on phase separation, we make further simplifying assumptions. It is assumed that the t2g core spins are aligned ferromagnetically and that Jh-> 00 this effectively projects out or b electron spin opposite to that of the t2g core spins—we obtain an effectively spinless model. The above considerations lead us to the following extended b Hamiltonian... 

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