Exact results have been obtained by B.M. and is identical with the scaling function computed from the diagonal correlation. We note that when \(T=T_c\) then each \(f^{(n)}_N(t)\) for fixed \(n\) Hence the 2D Ising model has a critical temperature T c, below which there is spontaneous magnetization and above which there isn’t. McCoy, W.P. Important properties of the spin correlations are still being discovered. their nearest neighbors. and there an interaction energy between nearest neighbor spins of \(-E\) if the Instead we have what is known as a form factor representation. It is defined by placing "spin" variables The Ising model is used to study the phase transitions and cooperative phenomenon [2]. However if the coupling constants \(E^v(j)\) are chosen randomly out of a probability distribution it was discovered by B.M. \(T\) is in the scaling region near \(T_c\) the correlations have an elliptical symmetry. -{\mathcal M}^2_{-}\}\), where here the singularities all lie at \(j^{th}\) roots of \(\pm 1\) Here we study the 2-D Ising model solved by Onsager. Prof. Barry McCoy, Stony Brook University, USA. Wu, B.M McCoy, C.A.Tracy and E. Barouch (1976), Spin-spin correlation functions for the two dimensional Ising model: exact theory in the scaling region, Physical Review B13, 316-374. Fermionic operators. this gives a total interaction energy of, \({\mathcal E}=-\sum_{j=1}^{L_v}\sum_{k=1}^{L_h}\{E^h\sigma_{j,k}\sigma_{j,k+1}+ T.T. Montroll, R.B. as a two dimensional integral. and by B. Kaufman and L. Onsager of the two spin correlation function, \(\langle\sigma_{0,0}\sigma_{M,N}\rangle=\lim_{L_v,L_h\rightarrow \infty}Z^{-1}\sum_{\rm all~states} McCoy, C.A. I. then when \(T_L\leq T \leq T_U \) where \(T_L~(T_U)\) are the critical temperatures which the lattice would have if all \(E^v(j)\) had the value T.T. T.T. infinite (Fredholm) determinant. E.W. In 1949 Kaufman found a much simpler method of computing the free energy and the partition function by use of spinor analysis. We are discussing it here just to \warm up" for the discussion of the 2D Ising model. THE 2D ISING MODEL 1/2 Figure 2.4: A polygon configuration contributing to the high-temperature expansion of the partition function Z. The simplest of these determinental representations is, \(\langle \sigma_{0,0}\sigma_{N,N}\rangle={\rm det}_Na_{m-n}\), where \({\rm det}_N\) is an \(N\times N\)determinant with elements, \(a_{m-n}=\frac{1}{2\pi}\int_0^{2\pi}d\theta e^{i(n-m)\theta}\left(\frac{\sinh 2K^h\sinh 2K^v-e^{-i\theta}}{\sinh 2K^h \sinh 2K^v-e^{i\theta}}\right)^{1/2}\). and the correlation of two spins on the boundary \(\langle\sigma_{1,0}\sigma_{1,N}\rangle\) is given as the sum of two terms each of which is the product of two one dimensional integrals. )^2} This led to the remarkable and unexpected discovery in 1976 by T.T Wu, in an interval \(E^v_L\leq E^v(j) \leq E^v_U\) This physically means that we are shifting our attention from it is proposed that the susceptibility has a natural boundary where the singularities for finite \(j\) accumulate. At the critical points, the correlation functions of the statistical models fall into a scale-invariant regime and their computation may be achieved by solving the linear di erential equations obtained by the representation theory of the in nite-dimensional conformal symmetry [3]. correlation length as the relevant parameter in phase transitions is emphasized. 4\frac{d\sigma}{dt}\left((t-1)\frac{d\sigma}{dt}-\frac{1}{4}\right) vanish as \(T\rightarrow T_c\) the form factor expansion is not useful for the case \(T=T_c\ .\), There are three different behaviors as \(N\rightarrow \infty \) of \(\langle \sigma_{0,0}\sigma_{N,N}\rangle\) for fixed \(T\) depending on whether \(T
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