By S.V.G. Menon
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Having established the equivalence of the lattice gas model (for a simple fluid or a binary mixture) with the Ising model, it is now appropriate to consider modifications of the latter. 3 n - Vector Spin Models Earlier it was mentioned that for some examples of critical phenomena, the order parameter should have several components. Thus for planar ferromagnets, each spin variable is a two dimensional vector si . The component s1i (or s2i ) varies continuously between −1 to +1. This lattice model, with a twocomponent (n = 2) order parameter, is generally called the X − Y model.
And hence the theory shows universality. The spatial dimension (d = 4) above which Landau’s theory is consistent is known as the upper critical dimension. For n-vector models, two dimension is called the lower critical dimension since there is no sponteneous magnetization for d ≤ 2. For Ising model, the lower critical dimension is one. References 1. S. Ma, ‘Modern Theory of Critical Phenomena’, (Benjamin, Reading, Massachusetts, 1976). 2. M. E. Fisher, ‘Scaling, Universality and Renormalization Group Theory’, Lecture Notes in Physics, (Springer - Verlag), Vol.
V Using the expression obtained in the previous section for χ, when T < Tc , one gets 1 1 < s (x)s (0) > dx. = 4T a2 (Tc − T ) T V 48 Renormalization Group Theory The spin fluctuations have a correlation length ξ (for T < Tc ) and for x within ξ , < s (x)s (0) > may be taken to be nearly constant. For x larger than ξ , the correlation function is negligible. So, the above relation can be approximated as 1 ≈< s 2 > ξ d . 4a2 (Tc − T ) Therefore, the mean square fluctuation in Gaussian approximation is ≈ ξ −d 1 c = 4a2 (Tc − T ) 4a2 2a2 −d/2 (Tc − T )d/2−1 .