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Physical Review

Correlations along a Line in the Two-Dimensional Ising Model

Leo P. Kadanoff

Published Dec 10, 1969• 51 citations
Publisher

Abstract

The $2n$-spin correlation function for the two-dimensional Ising model at $T={T}_{c}$ is evaluated for the special case in which all the spins lie along a straight line, separated by many lattice constants. The resulting $2n$-spin function is simply a quotient of products of two-spin correlations. A hypothesis of reducibility of fluctuations in the critical state is introduced. This hypothesis asserts that the product of any two local fluctuating quantities in the same neighborhood of space may be effectively replaced by a finite sum of local fluctuating quantities in this neighborhood. As a result, the previously found form for the $2n$-spin function may be used to evaluate the correlation function of $n$ energy densities when all $n$ points lie on a line. The $n$-energy correlation function is simply a sum of products of two-energy correlations. The quotient form for the spin correlation plus scaling is shown to immediately imply the logarithmic specific heat.

DOI
10.1103/physrev.188.859

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The energy-density---energy-density and energy-density---spin correlation functions are evaluated for the two-dimensional square Ising model. Results are obtained at zero magnetic field and near the critical point, i.e., for $\ensuremath{\epsilon}=(\frac{4J}{\mathrm{kT}})|(T\ensuremath{-}\frac{{T}_{c}}{{T}_{c}})|\ensuremath{\ll}1.$ The correlation functions are ${〈{E}_{R}{E}_{0}〉\ensuremath{-}{〈E〉}^{2},={(\frac{2J}{\ensuremath{\pi}})}^{2}{\ensuremath{\epsilon}}^{2}[{{K}_{1}}^{2}(\ensuremath{\epsilon}R)\ensuremath{-}{{K}_{0}}^{2}(\ensuremath{\epsilon}R)],}{〈{E}_{R}{\ensuremath{\sigma}}_{0}〉\ensuremath{-}〈E〉〈\ensuremath{\sigma}〉,=(\frac{2J}{\ensuremath{\pi}})\ensuremath{\epsilon}〈\ensuremath{\sigma}〉\ensuremath{\int}{2\ensuremath{\epsilon}R}^{\ensuremath{\infty}}\mathrm{ds} {s}^{\ensuremath{-}2}{e}^{\ensuremath{-}8},}$ where RRR is the distance between the points in question and K0{K}_{0}K0​ and K1{K}_{1}K1​ are modified Bessel functions of the second kind. Very similar results hold if the coupling constants are different in the two directions. The above results agree in form with the structure predicted by the "scaling-law" approach.
The energy-energy correlation function of the two-dimensional Ising model with weakly fluctuating random bonds is evaluated in the large scale limit. Two correlation lengths exist in contrast to one correlation length in the pure 2D Ising model: one is finite whereas the other is divergent at the critical points. The corresponding exponent of the divergent correlation length is νe=1/2ν_e=1/2νe​=1/2 in contrast to the pure system where νe=1ν_e=1νe​=1. The calculation is based on a previously developed effective field theory for the energy density fluctuations.
The effects of random impurities on the various thermodynamic functions are studied with the aid of the two-dimensional Ising model. Of special interest are the limiting forms of these functions near the critical point. This problem is approached by a Green's function formulation as developed by Kadanoff. Since experiments only measure the average effects of impurities, we consider the average of the Green's function over all distributions of impurities and calculate it by perturbation theory. From this function we find the location of the new critical temperature and the temperature dependence of the coherence length. Thermodynamic and correlation functions are calculated by means of perturbation expansions and their functional dependence on the coherence length investigated. It is found that the magnetic properties such as the magnetization, susceptibility, and spin-spin correlations retain the same functional dependence on the coherence length. The specific heat is found to have a finite value at the critical point, and the T = Tc form of the energy-energy correlation c function is changed. The thermodynamic functions exhibit new critical indices very close to Tc. These renormalized exponents have been predicted c in literature and are found to satisfy the scaling laws predictions.
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The direct correlation function of the one-dimensional lattice-gas (Ising) model with nearest-neighbor and next-nearest-neighbor interactions is calculated exactly. It is shown that, depending on the strengths and signs of the coupling constants, the direct correlation function can either have a finite range equal to that of the interactions, or decay exponentially in a monotonic or oscillatory fashion. At the critical point, which occurs at zero temperature, the direct correlation function is found in all cases to have exactly the range of the interactions, while its values become unbounded; this is in contrast to earlier nonrigorous results. The second moment of the direct correlation function is found to diverge at the critical point, in the neighborhood of which it becomes proportional to the correlation length of the density (spin) fluctuations, confirming a prediction of scaling theory in one dimension.