1. the spontaneous magnetization defined as, \( {\mathcal M}_{-}=\lim_{H\rightarrow 0+}M(H)\), \({\mathcal M}(H)=\lim_{L_v,L_h\rightarrow \infty} Z^{-1}\sum_{ {\rm all~states} }\sigma_{1,1}e^{-{\mathcal E}/k_BT}\), The exact expression was announced by Onsager in 1949 and proven by C.N. is fixed but \(E^h/E^v\) is allowed to vary. \(\langle\sigma_{0,0}\sigma_{M,N}\rangle_{\pm}\ .\) 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, \(a_{m-n}=\frac{1}{2\pi}\int_0^{2\pi}d\theta e^{i(n-m)\theta}\left(\frac{\sinh 2K^h\sinh 2K^v-e^{-i\theta}}{\sinh 2K^h \sinh 2K^v-e^{i\theta}}\right)^{1/2}\). Important properties of the spin correlations are still being discovered. the last term is the interaction with an external magnetic field and the energies \(E^h\) and \(E^v\) are allowed to be different. Exact results may also be obtained if the coupling \(E^v\) T.T. Barry McCoy (2010), Scholarpedia, 5(7):10313. where \(K(t^{1/2})\) (and \(E(t^{1/2})\)) is the complete elliptic integral of the first (second) kind. However, it was found by B.M. Thus we have the susceptibility expansions, \(\chi_{-}(T)=\frac{ {\mathcal M}^2_{-} }{k_BT}\sum_{n=1}^{\infty}{\hat\chi}^{(2n)}(T)\), \(\chi_{+}(T)= comes from the large \(N\) behavior of \(f^{(2)}_N(t)\), \(\langle\sigma_{0,0}\sigma_{N,N}\rangle_{-}=(1-t)^{1/4}\left\{1+\frac{t^{N+1}}{\pi N^2(1-t)^2}+\cdots\right\}\), 3. t^{N+1}+\cdots\}\), with \(\lambda=1\) and \((a)_N=a(a+1)\cdots (a+N-1)\ .\), The general two spin correlation \(\langle\sigma_{0,0}\sigma_{M,N}\rangle\) Wu in 1967 for the properties of spins near the boundary of a half plane with a magnetic field \(H_b\) interacting with the row of boundary spins \(\sigma_{1,m}\ .\) The spontaneous magnetization of a spin on the boundary is, \({\mathcal M}_b=\left[\frac{\cosh 2K^v-\coth2K^h}{\cosh 2K^v-1}\right]^{1/2}\), Thus the critical exponent \(\beta=1/2\) for the boundary magnetization. As \(j\rightarrow \infty\) these singularities become dense and therefore This limit is referred to as the scaling limit. satisfies a nonlinear differential equation which could serve as an alternative characterization of the function. On a rectangular lattice of \(L_v\) rows and \(L_h\) columns There are at least five different methods which have been used to compute the free energy of the Ising model. In 1949 Kaufman found a much simpler method of computing the free energy and the partition function by use of spinor analysis. may be exactly calculated in the thermodynamic limit where \(L_v,L_h \rightarrow \infty\) \prod_{1\leq j < k \leq n}(x_{2j}-x_{2k})}{\prod_{1\leq j \leq n+1} Thus if we define, \(R^2=\left(\frac{\sinh2K^h_c}{\sinh2K^v_c}\right)^{1/2}M^2 +\left(\frac{\sinh2K^v_c}{\sinh2K^h_c}\right)^{1/2}N^2\), and define a scaling limit as \(T\rightarrow T_c\) and \(R\rightarrow \infty\) \(M.N\ .\), The remaining thermodynamic property to be computed is the magnetic susceptibility at \(H=0\) which is computed from, \(\chi_{\pm}(T)=\frac{1}{k_BT}\sum_{M=-\infty}^{\infty}\sum_{N=-\infty}^{\infty} W.P. I. as a two dimensional integral. This is the statistical mechanical analogue of mass renormalization in quantum field theory. In two dimensions this is usually called the square lattice, in three the cubic lattice and in one dimension it is often refered to as a chain. The leading behavior as \(N\rightarrow \infty\) is given by the large \(N\) behavior of \(f^{(1)}_N(t)\), \(\langle\sigma_{0,0}\sigma_{N,N}\rangle_{+}\sim(1-t)^{1/4}f^{(1)}_N(t)=\frac{t^{N/2}}{(\pi N)^{1/2}(1-t)^{1/4}}+\cdots\). -- The Onsager solution to the two-dimensional Ising model is phrased in the language of thermodynamic Green's functions. feature of the Ising model allows an exact microscopic description of the behavior near the critical temperature. \(\sinh 2E^v/k_BT\sinh2E^h/k_BT\) \(T>T_c\ .\), 1, The case T=Tc. correlation length as the relevant parameter in phase transitions is emphasized. For cases 2 and 3 we must have \( N \gg(1-t)^{-1}\) for the expansions to be valid. To make further progress it was necessary to invent mathematics to efficiently study L. P. KADANOFF (**) Departme~t of Physics, University of Illinois - Urbana, Ill. (ricevuto il 4 Gennaio 1966) Summary. Almost all of the properties presented above for the diagonal correlation hold is far less general than Onsager's algebra but is sufficiently powerful that it can also be used to compute the correlation functions in terms of determinants. However if the coupling constants \(E^v(j)\) are chosen randomly out of a probability distribution it was discovered by B.M. mathematical development, the full generality of which is still under development. 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. Perk that the general correlation function \(\langle \sigma_{0,0}\sigma_{M,N}\rangle\) does satisfy a quadratic difference equation in \(M^2+N^2\) is large. T.T. and \(K_0(2\theta)\) is the modified Bessel function. )^2 \pi^{2n}}\int_0^1\prod_{k=1}^{2n}x_k^Ndx_k\prod_{j=1}^n The terms \(t^{N}\) and \(t^{N/2}\) are often written as, \(t^{N/2}=e^{(N/2)\ln t}\) and \(t^N=e^{N\ln t}\), where \(\xi\) is called the correlation length. When \(N\) is small this is an efficient representation of the correlation. {\prod_{j=1}^{n+1} x_{2j-1}(1-x_{2j-1})(1-tx_{2j-1})}\right)^{1/2}\), \(\times\left(\frac{\prod_{1\leq j < k \leq n+1}(x_{2j-1}-x_{2k-1}) From this we find that at \(T=T_c\) \{\langle \sigma_{0,0}\sigma_{M,N}\rangle_{\pm}-{\mathcal M}^2_{-}\}\), by use of the form factor expansions for Kaufman computed the partition function \(Z\) as the sum of the Pfaffians (the square root of the determinant) of four antisymmetric matrices and each of these Pfaffians is evaluated as a double product. However, unlike the free energy and the spontaneous magnetization where the specific heat \(c\) diverges as \(T\rightarrow T_c\) as, and thus the critical exponent \(\alpha=0\ .\) for the diagonal susceptibility Commuting transfer matrices. We are discussing it here just to \warm up" for the discussion of the 2D Ising model. \), where \({\mathcal M}_+=(1-(\sinh 2 K^v \sinh 2K^h)^2)^{1/8}\ .\) The first of these determinants for spin correlations were found in 1949 by Kaufman and Onsager The amplitudes \(A_{\pm}\) are different for \(T\) above or below \(T_c\) and the ratio \(A_{+}/A_{-}\) is approximately \(12\pi\ .\) and in 1980 to the equally remarkable discovery by M. Jimbo and T.Miwa that for all temperatures the diagonal correlation function \(\langle \sigma_{0,0}\sigma_{N,N}\rangle\) is given in terms of a Painlevé function of the sixth kind. on a diagonal are equal, are called Toeplitz. the determinants of Kaufman and Onsager for the two spin correlation function when the separation \left(\frac{\prod_{j=1}^n x_{2j}(1-x_{2j})(1-tx_{2j})} their nearest neighbors. where \(\lambda=1\) 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 Wu in 1966 and has been developed in great length by several authors. McCoy, W.P. At the critical temperature \(T=T_c\) the determinant for the diagonal correlation may be reduced to a much more explicit expression, \(\langle\sigma_{0,0}\sigma_{N,N}\rangle|_{T=T_c} Wu, B.M McCoy, C.A.Tracy and E. Barouch (1976). Wu in 1968 that the specific heat is finite at the critical temperature and that the logarithmic singularity of the nonrandom lattice has become an infinitely differentiable essential singularity. Unlike the form factors themselves which only have singularities at \(T=T_c\) the \({\hat\chi}^{(j)}(T)\) have many singularities in the complex temperature The Hamiltonian, H of the Ising model is give by: *= − , Í O Ü O Ý Ü. Ý J ij is the interaction energy between spins at lattice point i and j. where \(\epsilon\) is the distance of \(T\) from the singularity. 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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