# Ising model: exact results

Post-publication activity

Curator: Barry McCoy

The Ising model is a simple classical model of a ferromagnet which has the remarkable property that in two dimensions its physical properties may be exactly calculated. These exact calculations have given microscopic insight into the many body collective phenomena of phase transitions and have developed new areas of mathematics.

## Historical Introduction

The Ising model was introduced by Lenz in 1920 and solved in one dimension by Ising in 1925. It is defined by placing "spin" variables $$\sigma$$ which take on the values $$\pm 1$$ on the sites of a lattice and there an interaction energy between nearest neighbor spins of $$-E$$ if the spins have the same value and $$+E$$ if the spins have opposite values. On a rectangular lattice of $$L_v$$ rows and $$L_h$$ columns with periodic boundary conditions in both directions this gives a total interaction energy of

$${\mathcal E}=-\sum_{j=1}^{L_v}\sum_{k=1}^{L_h}\{E^h\sigma_{j,k}\sigma_{j,k+1}+ E^v\sigma_{j,k}\sigma_{j+1,k}+H\sigma_{j,k}\}$$

where $$\sigma_{j,k}$$ is the spin in row $$j$$ and column $$k\ ,$$ the last term is the interaction with an external magnetic field and the energies $$E^h$$ and $$E^v$$ are allowed to be different.

In 1944 L. Onsager made one of the most important discoveries in theoretical physics of the 20th century by finding that when $$H=0$$ the free energy and the specific heat of the Ising model may be exactly calculated in the thermodynamic limit where $$L_v,L_h \rightarrow \infty$$ as a two dimensional integral. From this integral Onsager found that there is a temperature $$T_c$$ defined by

$$\sinh2E^h/k_BT_c\sinh2E^v/k_BT_c=1$$

with $$k_B$$ being Boltzmann's constant, where the specific heat $$c$$ diverges as $$T\rightarrow T_c$$ as

$$c\sim A_C\ln|T-T_c|$$

and thus the critical exponent $$\alpha=0\ .$$ This initial discovery was followed in 1949 by the calculation by B. Kaufman of the partition function

$$Z=\sum_{ {\rm all~states} }e^{-{\mathcal E}/k_BT}$$

for the finite lattice and by B. Kaufman and L. Onsager of the two spin correlation function

$$\langle\sigma_{0,0}\sigma_{M,N}\rangle=\lim_{L_v,L_h\rightarrow \infty}Z^{-1}\sum_{\rm all~states} \sigma_{0,0}\sigma_{M,N}e^{-{\mathcal E}/k_BT}$$

in terms of determinants which for small separations between the spins may be easily evaluated,

The next property to be studied was the spontaneous magnetization defined as

$${\mathcal M}_{-}=\lim_{H\rightarrow 0+}M(H)$$

where

$${\mathcal M}(H)=\lim_{L_v,L_h\rightarrow \infty} Z^{-1}\sum_{ {\rm all~states} }\sigma_{1,1}e^{-{\mathcal E}/k_BT}$$

The exact expression was announced by Onsager in 1949 and proven by C.N. Yang in 1952. It is nonzero only for $$T<T_c$$ and vanishes as $$T\rightarrow T_c-$$ as

$${\mathcal M}_{-}\sim A_M(T_c-T)^{1/8}$$

and thus the critical exponent $$\beta=1/8\ .$$

To make further progress it was necessary to invent mathematics to efficiently study the determinants of Kaufman and Onsager for the two spin correlation function when the separation $$M^2+N^2$$ is large. This new mathematics was initiated by T.T. Wu in 1966 and has been developed in great length by several authors. This led to the remarkable and unexpected discovery in 1976 by T.T Wu, B.M. McCoy, C.A. Tracy and E. Barouch that for temperatures near $$T_c$$ the correlation can be expressed in terms of a Painlevé function of the third kind, 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. Important properties of the spin correlations are still being discovered.

The final thermodynamic property of interest is the magnetic susceptibility at $$H=0$$

$$\chi(T)=d{\mathcal M}(H)/dH|_{H=0}$$

which is expressed in terms of $$\langle\sigma_{0,0}\sigma_{M,N}\rangle$$ as

$$\chi(T)=\frac{1}{k_BT}\sum_{M=-\infty}^{\infty}\sum_{N=-\infty}^{\infty}\{\langle\sigma_{0,0} \sigma_{M,N}\rangle-{\mathcal M}_{-}^2\}$$

When $$T\rightarrow T_c$$ the susceptibility diverges as

$$\chi(T)\sim A_{\pm}|T-T_c|^{-7/4}$$

and thus the critical exponent $$\gamma=7/4\ .$$ The amplitudes $$A_{\pm}$$ are different for $$T$$ above or below $$T_c$$ and the ratio $$A_{+}/A_{-}$$ is approximately $$12\pi\ .$$ However, unlike the free energy and the spontaneous magnetization the full mathematical structure of the susceptibility as a function of temperature is as yet unknown.

## Methods of Solution

There are at least five different methods which have been used to compute the free energy of the Ising model

1. Onsager's algebra. Onsager's 1944 computation computes the free energy by discovering that the model can be described by what is now known as an $$sl_2$$ loop algebra. This is an extremely deep mathematical development, the full generality of which is still under development.

2. Fermionic operators. In 1949 Kaufman found a much simpler method of computing the free energy and the partition function by use of spinor analysis. This was substantially simplified in 1964 by Schultz, Mattis and Lieb who reduced the spinor analysis of the partition function to the evaluation of an exponential of a quadratic form in Fermion creation and annihilation operators which can be diagonalized by linear algebra. This method 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.

3. Combinatoric. A related method which reduces the computation of the partition function and correlation functions to a graph counting problem was given by Kac, Ward, Potts, Hurst and Green in several papers, and the relation to a solvable problem in dimer statistics was presented by P.W. Kastelyn in 1963.

4. Commuting transfer matrices. An extremely powerful method which has been generalized to solve many classes of models in 2 dimensions was invented by R.J. Baxter which relies on an invariance exhibited by the anisotropic model where $$\sinh 2E^v/k_BT\sinh2E^h/k_BT$$ is fixed but $$E^h/E^v$$ is allowed to vary.

5.The 399th solution. A completely new method found by R.J. Baxter and I.G. Enting in 1978.

## Partition Function and Specific Heat

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. In the thermodynamic limit where $$L_h,~L_v\rightarrow \infty$$ these four terms are equal to leading order, with the result that the free energy per site is

$$-F/k_BT=\lim_{L_v,L_h\rightarrow \infty}(L_vL_h)^{-1}\ln Z$$ $$=\ln2+\frac{1}{8\pi^2}\int_0^{2\pi}d\theta_1\int_0^{2\pi}d\theta_2 \ln[\cosh2K^h\cosh2K^v-\sinh 2K^h \cos \theta_1-\sinh 2K^v\cos \theta_2]$$

where $$K^{(v,h)}=E^{(v,h)}/k_BT$$

The specific heat $$c$$ is obtained from the free energy as $$c=-Td^2F/dT^2$$

## Spontaneous Magnetization

The exact value of the spontaneous magnetization is

$${\mathcal M}_{-}=\{1-(\sinh 2E^v/k_BT \sinh 2E^h/k_BT)^{-2}\}^{1/8}$$

## Spin Correlation Functions

The Ising model is unique among all problems in statistical because not only can the macroscopic thermodynamic properties of free energy and spontaneeous magnetization be exactly computed, but the spin correlation functions are exactly computable also. This unique feature of the Ising model allows an exact microscopic description of the behavior near the critical temperature. The phenomenology of scaling theory originates in these computations.

Our ability to carry out these exact computations originates in the fact that all spin correlations may be represented as determinants. There are, in fact many different equivalent representations in terms of determinants of different sizes and even representations in terms of infinite (Fredholm) determinants have proven to be most useful.

The first of these determinants for spin correlations were found in 1949 by Kaufman and Onsager and in 1963 by E.W. Montroll, R.B. Potts and J.C. Ward. The size of these determinants grows with the separation between the spins. The study of the spin-spin correlations for large separations was initiated in 1966 by T.T. Wu for row correlations and culminated in 1976 with the work of Wu, McCoy, Tracy and Barouch for the general case $$\langle \sigma_{0,0}\sigma_{M,N}\rangle\ .$$ The presentation of these results in full generality is made somewhat tedious because of their angular dependence and in order to keep the notation to a minimum we will here present in detail the results only for diagonal correlations $$\langle\sigma_{0,0}\sigma_{N,N}\rangle$$ in detail.

### Determinental Representation

The two spin correlation function on the diagonal $$\langle \sigma_{0,0}\sigma_{N,N}\rangle$$ can be written as a determinant in many different ways. 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

$$a_{m-n}=\frac{1}{2\pi}\int_0^{2\pi}d\theta e^{i(n-m)\theta}\left(\frac{\sinh 2K^h\sinh 2K^v-e^{-i\theta}}{\sinh 2K^h \sinh 2K^v-e^{i\theta}}\right)^{1/2}$$

where $$1\leq m,n\leq N\ .$$ Determinants with this property, that all elements on a diagonal are equal, are called Toeplitz. When $$N$$ is small this is an efficient representation of the correlation.

### Diagonal Correlation at T=Tc

At the critical temperature $$T=T_c$$ the determinant for the diagonal correlation may be reduced to a much more explicit expression

$$\langle\sigma_{0,0}\sigma_{N,N}\rangle|_{T=T_c} =\left(\frac{2}{\pi}\right)^N\prod_{l=1}^{N-1}[1-\frac{1}{4l^2}]^{l-N}$$

### Form Factor Representations

When $$T\neq T_c$$ there is no simple product representation of the correlation function. Instead we have what is known as a form factor representation. For $$T<T_c$$ this representation is

$$\langle\sigma_{0,0}\sigma_{N,N}\rangle_{-}={\mathcal M}_{-}^2[1+\sum_{n=1}^{\infty}f^{(2n)}_{N}(t)]$$

with $$t=(\sinh 2K^v \sinh 2K^h)^{-2}$$ and

$$f^{(2n)}_N(t)=\frac{t^{n(N+n)}}{(n!)^2 \pi^{2n}}\int_0^1\prod_{k=1}^{2n}x_k^Ndx_k\prod_{j=1}^n \left(\frac{x_{2j-1}(1-x_{2j})(1-tx_{2j})}{x_{2j}(1-x_{2j-1})(1-tx_{2j-1})}\right)^{1/2}$$ $$\times\left(\frac{\prod_{1\leq j < k\leq n}(x_{2j-1}-x_{2k-1})(x_{2j}-x_{2k})} {\prod_{1\leq j \leq n}\prod_{1\leq k \leq n}(1-tx_{2j-1}x_{2k})}\right)^2$$

and for $$T>T_c$$

$$\langle\sigma_{0,0}\sigma_{N,N}\rangle_{+}={\mathcal M}_+^2\sum_{n=0}^{\infty}f^{(2n+1)}_{N}(t)$$

with $$t=(\sinh 2K^v \sinh 2K^h)^2$$ and

$$f^{(2n+1)}_N(t)=\frac{t^{(2n+1)N/2+n(n+1)}}{n!(n+1)!\pi^{2n+1}} \int_0^1\prod_{k=1}^{2n+1}x_k^Ndx_k \left(\frac{\prod_{j=1}^n x_{2j}(1-x_{2j})(1-tx_{2j})} {\prod_{j=1}^{n+1} x_{2j-1}(1-x_{2j-1})(1-tx_{2j-1})}\right)^{1/2}$$

$$\times\left(\frac{\prod_{1\leq j < k \leq n+1}(x_{2j-1}-x_{2k-1}) \prod_{1\leq j < k \leq n}(x_{2j}-x_{2k})}{\prod_{1\leq j \leq n+1} \prod_{1\leq k \leq n}(1-tx_{2j-1}x_{2k})}\right)^2$$

where $${\mathcal M}_+=(1-(\sinh 2 K^v \sinh 2K^h)^2)^{1/8}\ .$$ This representation of the form factor was derived in 2007 by I. Lyberg and B.M.McCoy. We note that when $$T=T_c$$ then each $$f^{(n)}_N(t)$$ for fixed $$n$$ is finite and thus because both $${\mathcal M}_{\pm}$$ vanish as $$T\rightarrow T_c$$ the form factor expansion is not useful for the case $$T=T_c\ .$$

### Large Separation Behavior

There are three different behaviors as $$N\rightarrow \infty$$ of $$\langle \sigma_{0,0}\sigma_{N,N}\rangle$$ for fixed $$T$$ depending on whether $$T<T_c,~T=T_c$$ or $$T>T_c\ .$$

1, The case T=Tc. We find from the product representation that

$$\langle \sigma_{0,0}\sigma_{N,N}\rangle=AN^{-1/4}\left(1-\frac{1}{64N^2}+O(N^{-4})\right)$$

$$A=2^{1/12}\exp[3\zeta'(-1)]\sim 0.6450\cdots$$

and $$\zeta'(-1)$$ is the derivative of the Riemann zeta function at $$-1\ .$$

2, The case T<Tc . We find that the leading approach to $${\mathcal M}_{-}^2$$ 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. The case T>Tc. The leading behavior as $$N\rightarrow \infty$$ is given by the large $$N$$ behavior of $$f^{(1)}_N(t)$$

$$\langle\sigma_{0,0}\sigma_{N,N}\rangle_{+}\sim(1-t)^{1/4}f^{(1)}_N(t)=\frac{t^{N/2}}{(\pi N)^{1/2}(1-t)^{1/4}}+\cdots$$

For cases 2 and 3 we must have $$N \gg(1-t)^{-1}$$ for the expansions to be valid.

The terms $$t^{N}$$ and $$t^{N/2}$$ are often written as

$$t^{N/2}=e^{(N/2)\ln t}$$ and $$t^N=e^{N\ln t}$$

and $$\ln t$$ is denoted by

$$\ln t=\xi^{-1}$$

where $$\xi$$ is called the correlation length. As $$t\rightarrow 1$$ (i.e. $$T\rightarrow T_c$$) this correlation length diverges as

$$\xi \sim (1-t)^{-1}$$

### Scaling Limit

The large $$N$$ behaviors of the two point function for fixed $$T\neq T_c$$ do not smoothly connect to the large $$N$$ behavior at $$T=T_c$$ because the condition $$N\gg(1-t)^{-1}$$ is violated. To obtain an interpolating function between the regimes we need to consider the case where

$$N(1-t)=r$$

is fixed in the limit

$$N\rightarrow \infty$$ and $$t\rightarrow 1\ .$$

This limit is referred to as the scaling limit. The physical meaning is that we have shifted our attention from the scale of the lattice spacing to the scale of the correlation length. This is the statistical mechanical analogue of mass renormalization in quantum field theory.

In this scaling limit we define scaling functions for $$T$$ above and below $$T_c$$ as

$$G_{\pm}(r)=\lim_{ {\rm scaling} }{\mathcal M}_{\pm}^{-2}\langle\sigma_{0,0}\sigma_{N,N}\rangle_{\pm}$$

This physically means that we are shifting our attention from the scale of definition of $$\sigma=\pm 1$$ to the scale of $${\mathcal M}_{\pm}$$near $$T_c\ .$$ This is the analogue of wave function renormalization in quantum field theory and $$G_{\pm}(r)$$ is the analogue of the Greens function.

### Painlevé Representation

The scaling functions $$G_{\pm}(r)$$ have the remarkable property announced in 1973 with a derivation published in 1976 by Wu, McCoy, Tracy and Barouch that

$$G_{\pm}(r)=\frac{1}{2}[\eta(r/2)^{-1/2}\mp\eta(r/2)^{1/2}] \exp\int_{r/2}^{\infty}d\theta\frac{\theta}{4\eta^2}[(1-\eta^2)^2-(\eta')^2]$$

where $$\eta(\theta)$$ is a Painlevé function of the third kind defined by

$$\frac{d^2\eta}{d\theta^2}= \frac{1}{\eta}\left(\frac{d\eta}{d\theta}\right)^2-\frac{1}{\theta}\frac{d\eta}{d\theta} +\eta^3-\eta^{-1}$$

with the boundary condition as $$\theta\rightarrow\infty$$

$$\eta(\theta)\sim 1-\lambda\frac{2}{\pi}K_0(2\theta)$$

where $$\lambda=1$$ and $$K_0(2\theta)$$ is the modified Bessel function.

In 1980 a corresponding characterization of the diagonal correlation function for all temperatures in terms of a Painlevé VI function was found by M. Jimbo and T. Miwa

$$\left(t(t-1)\frac{d^2\sigma}{dt^2}\right)^2 =N^2\left((t-1)\frac{d\sigma}{dt}-\sigma\right)^2- 4\frac{d\sigma}{dt}\left((t-1)\frac{d\sigma}{dt}-\frac{1}{4}\right) \left(t\frac{d\sigma}{dt}-\sigma\right)$$

where for $$T<T_c$$

$$\sigma_{-}=t(t-1)\frac{d\ln\langle\sigma_{0,0}\sigma_{N,N}\rangle_{-}}{dt}-\frac{t}{4}$$

and for $$T>T_c$$

$$\sigma_{+}=t(t-1)\frac{d\ln\langle\sigma_{0,0}\sigma_{N,N}\rangle_{+}}{dt}-\frac{1}{4}$$

with the boundary conditions specified at $$t=0$$ to agree with the form factor expansions at $$t=0\ .$$ For $$T<T_c$$ this is

$$\langle\sigma_{0,0}\sigma_{N,N}\rangle=(1-t)^{1/4}\{1+\lambda^2\frac{(1/2)_N(3/2)_N}{4((N+1)!)^2} t^{N+1}+\cdots\}$$

with $$\lambda=1$$ and $$(a)_N=a(a+1)\cdots (a+N-1)\ .$$

### Two Spin Correlation in General Position

The general two spin correlation $$\langle\sigma_{0,0}\sigma_{M,N}\rangle$$ is studied most efficiently by representing the correlation as an infinite (Fredholm) determinant. This is the method used in 1976 by Wu, McCoy, Tracy and Barouch. Almost all of the properties presented above for the diagonal correlation hold for the general case once appropriate angle dependencies are included. In particular when $$T$$ is in the scaling region near $$T_c$$ the correlations have an elliptical symmetry. 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 $$R\rightarrow \infty$$ with

$$r=|T-T_c|R$$

fixed then the scaling function

$$G_{\pm}(r)=\lim_{ {\rm scaling} }{\mathcal M}_{\pm}^{-2}\langle \sigma_{0,0}\sigma_{M,N}\rangle_{\pm}$$

depends only on $$r$$ and not on the ratio $$\sinh 2K^v_c/\sinh2K^h_c$$ and is identical with the scaling function computed from the diagonal correlation.

The only property of the diagonal correlation which has not been extended to the general case is the representation as a Painlevé VI function. However, it was found by B.M. McCoy and T.T. Wu and by J.H.H. Perk that the general correlation function $$\langle \sigma_{0,0}\sigma_{M,N}\rangle$$ does satisfy a quadratic difference equation in $$M.N\ .$$

## Magnetic Susceptibility

The remaining thermodynamic property to be computed is the magnetic susceptibility at $$H=0$$ which is computed from

$$\chi_{\pm}(T)=\frac{1}{k_BT}\sum_{M=-\infty}^{\infty}\sum_{N=-\infty}^{\infty} \{\langle \sigma_{0,0}\sigma_{M,N}\rangle_{\pm}-{\mathcal M}^2_{-}\}$$

by use of the form factor expansions for $$\langle\sigma_{0,0}\sigma_{M,N}\rangle_{\pm}\ .$$ Thus we have the susceptibility expansions

$$\chi_{-}(T)=\frac{ {\mathcal M}^2_{-} }{k_BT}\sum_{n=1}^{\infty}{\hat\chi}^{(2n)}(T)$$

$$\chi_{+}(T)= \frac{ {\mathcal M}^2_{+} }{k_BT}\sum_{n=0}^{\infty}{\hat\chi}^{(2n+1)}(T)$$

where the explicit expressions for $${\hat\chi}^{(j)}(T)$$ for general $$j$$were first given as $$j$$ fold integrals by B.G. Nickel. For $$j=1,2$$ these integrals may be explicitly evaluated and in the case $$E^v=E^h$$ they reduce to

$${\hat \chi}^{(1)}(T)=\frac{1}{(1-t^{1/2})^2}$$

$${\hat \chi}^{(2)}(T)=\frac{(1+t)E(t^{1/2})-(1-t)K(t^{1/2})}{3\pi(1-t^{1/2})(1-t)}$$

where $$K(t^{1/2})$$ (and $$E(t^{1/2})$$) is the complete elliptic integral of the first (second) kind.

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 plane located at

$$\cosh 2E^v/k_BT\cosh 2E^h/k_BT-\sinh 2E^h/k_BT\cos(2\pi m'/j)-\sinh 2E^v/k_BT\cos (2\pi m/j)=0$$

with $$1\leq m,m'\leq j\ .$$

For $$T>T_c$$ the singularity in $${\hat\chi}^{(2n+1)}(T)$$ is of the form

$$A_{2n+1}\epsilon^{2n(n+1)-1}\ln \epsilon$$

and for $$T<T_c$$ the singularity in $${\hat\chi}^{(2n)}(T)$$ is of the form

$$A_{2n}\epsilon^{2n^2-3/2}$$

where $$\epsilon$$ is the distance of $$T$$ from the singularity.

A similar phenomenon was found by S. Boukra et al. for the diagonal susceptibility defined by

$$\chi_d(T)=\frac{1}{k_BT}\sum_{N=-\infty}^{\infty}\{\langle \sigma_{0,0}\sigma_{N,N}\rangle -{\mathcal M}^2_{-}\}$$

where here the singularities all lie at $$j^{th}$$ roots of $$\pm 1$$ on the unit circle $$|t|=1\ .$$

As $$j\rightarrow \infty$$ these singularities become dense and therefore it is proposed that the susceptibility has a natural boundary where the singularities for finite $$j$$ accumulate. Such a natural boundary is a new and unexpected feature, and the search for a rigorous demonstration of its existence and an interpretation of its meaning is still under investigation. It is of particular interest to determine if the susceptibility satisfies a nonlinear differential equation which could serve as an alternative characterization of the function.

## Boundary Properties

Exact results have been obtained by B.M. McCoy and T.T. 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. The boundary magnetization as a function of the boundary field is given as a single integral 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. From this we find that at $$T=T_c$$ the boundary two spin vanishes as $$1/N$$ as $$N\rightarrow \infty$$ when $$H_b=0$$ and approaches $${\mathcal M}_b^2(H_b)$$ as $$1/N^4$$ when $$H_b\neq 0\ .$$

## Random Layered Ising Model

Exact results may also be obtained if the coupling $$E^v$$ between rows $$j$$ and $$j+1$$ is allowed to depend on $$j\ .$$ If the coupling constants have a periodicity $$E^v(j+J)=E^v(j)$$ then the specific heat still has a logarithmic divergence but the amplitude is reduced from the non layered case solved by Onsager. However if the coupling constants $$E^v(j)$$ are chosen randomly out of a probability distribution it was discovered by B.M. McCoy and T.T. 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.

We conclude by noting that if the random coupling constants are restricted to lie in an interval $$E^v_L\leq E^v(j) \leq E^v_U$$ 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 $$E^v_L~(E^v_U)$$ that the correlations in a row will decay algebraically instead of exponentially with a power decay which depends on $$T\ .$$ The decay is exponential only if $$T$$ lies outside this range.