Ising fixed point

Infinite-Randomness Quantum Ising Critical Fixed Points. 

For 4fourth order terms are irrelevant in Eq. (78). An anisotropic (/i -renor-malized theory was constructed [85]. The Gaussian and Ising fixed point are unstable, as expected. A stable non-trivial fixed point exists and anisotropic critical exponents were calculated in 6-f dimensions. 

It should be realized that unlike the study of equilibrium thermodynamics for which a model is often mapped onto Ising system, elementary mechanism of atomic motion plays a deterministic role in the kinetic study. In an actual alloy system, diffusion of an atomic species is mainly driven by vacancy mechanism. The incorporation of the vacancy mechanism into PPM formalism, however, is not readily achieved, since the abundant freedom of microscopic path of atomic movement demands intractable number of variational parameters. The present study is, therefore, limited to a simple spin kinetics, known as Glauber dynamics [14] for which flipping events at fixed lattice points drive the phase transition. Hence, the present study for a spin system is regarded as a precursor to an alloy kinetics. The limitation of the model is critically examined and pointed out in the subsequent sections. 

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,... 

Let us now look into the case when there is a solubility loop and focus our attention on the behavior near the lower critical point with temperature c. For the usual Ising model the critical temperature (Tc=l//3c) is dependent on the lattice geometry and the model interactions, i.e., exp(-23cJ)=Kc, with Kc fixed by the lattice geometry, determines I3c. Since in our model, we have from (4) and (5) that JO)=-ln K(, ep)/2, it follows that... 

From Eqs. (24)-(27) we have seen that the standard mean-field power laws describing the singularities near the critical unmixing point can hold only for f = (jf//crit - 1) 1. Of course, when / -> xait at fixed N, we do expect that mean-field theory breaks down due to the neglect of thermal fluctuations, and in reality a crossover to the critical behavior described by the Ising model universality class [34, 35, 36] sets in. Thus, very close to the critical point, we expect the following critical exponents [35, 36]. 

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