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Physical Review Letters

Energy-Density Correlation Functions in the Two-Dimensional Ising Model with a Line Defect

Lee-Fen Ko, Helen Au-Yang, Jacques H. H. Perk

Published Mar 18, 1985• 28 citations
Publisher

Abstract

We consider the two-dimensional Ising model on a rectangular lattice with one row of horizontal coupling strengths changed. We shall present our results for correlation functions with two energy-density operators in arbitrary positions, restricting ourselves here to the scaling limit. These exact results show an unexpected complexity.
DOI
10.1103/physrevlett.54.1091

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An Ising model on a square lattice is studied, where one row of horizontal bonds has an energy ${{E}_{1}}^{\ensuremath{'}}$ different from all other horizontal bonds. The correlation of two spins is calculated in this row, resulting in exponents $\ensuremath{\beta}$ and $\ensuremath{\eta}$ which depend on ${{E}_{1}}^{\ensuremath{'}}$. The long-distance behavior of the correlation for fixed $T\ensuremath{\ne}{T}_{c}$ is found to have different forms depending upon the value of ${{E}_{1}}^{\ensuremath{'}}$.
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The 2n2n2n-spin correlation function for the two-dimensional Ising model at T=TcT={T}_{c}T=Tc​ is evaluated for the special case in which all the spins lie along a straight line, separated by many lattice constants. The resulting 2n2n2n-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 2n2n2n-spin function may be used to evaluate the correlation function of nnn energy densities when all nnn points lie on a line. The nnn-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.
Expressions for many-site correlation functions of a square-lattice Ising model have been derived for the case in which all sites lie in the same row. The results involve expressions for minors of Toeplitz matrices which are obtained by the Wiener-Hopf technique. We derive an expression for the magnetization at any site in a row if a finite number of defect bonds are distributed at given positions in that row; this expression can be used to study the effect of a dislocation. Some remarks about the decoupling of correlation functions are made.