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Physical review. B, Condensed matter

Four-spin correlation function of two parallel nonaligned pairs in the planar Ising model

Helen Au-Yang

Published May 1, 1981• 0 citations
Publisher

Abstract

The four-spin correlation function of two parallel nonaligned pairs in the two-dimensional rectangular Ising lattice is found to have the scaling form $〈{\ensuremath{\sigma}}_{00}{\ensuremath{\sigma}}_{\mathrm{NN}}{\ensuremath{\sigma}}_{J,J+l}{\ensuremath{\sigma}}_{J+L,J+L+l}〉\ensuremath{\sim}〈{\ensuremath{\sigma}}_{00}{\ensuremath{\sigma}}_{\mathrm{NN}}〉〈{\ensuremath{\sigma}}_{00}{\ensuremath{\sigma}}_{\mathrm{LL}}〉+{〈{\ensuremath{\sigma}}_{00}{\ensuremath{\sigma}}_{\mathrm{NN}}〉}^{c}{〈{\ensuremath{\sigma}}_{00}{\ensuremath{\sigma}}_{\mathrm{LL}}〉}^{c}{f}_{4}({x}_{1},{x}_{2},X,\ensuremath{\varphi})$, where ${f}_{4}$ is the scaling function. The scaling variables ${x}_{1},{x}_{2}$, and $X$ are, respectively, the lengths of the pairs ($N$ and $L$) and distance between the centers of the pairs scaled by the corresponding correlation lengths. The angle $\ensuremath{\varphi}$ measures the deviation from the diagonal axis of the line joining the centers. The scaling function is given as a series of ${x}_{1}$ and ${x}_{2}$, truncated at the order of ${({x}_{1}{x}_{2})}^{3}$.

DOI
10.1103/physrevb.23.4749

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The four-spin correlation function of two parallel nonaligned pairs in the two-dimensional rectangular Ising lattice is found to have the scaling form 〈\ensuremathσ00\ensuremathσNN\ensuremathσJ,J+l\ensuremathσJ+L,J+L+l〉\ensuremath∼〈\ensuremathσ00\ensuremathσNN〉〈\ensuremathσ00\ensuremathσLL〉+〈\ensuremathσ00\ensuremathσNN〉c〈\ensuremathσ00\ensuremathσLL〉cf4(x1,x2,X,\ensuremathφ)〈{\ensuremath{\sigma}}_{00}{\ensuremath{\sigma}}_{\mathrm{NN}}{\ensuremath{\sigma}}_{J,J+l}{\ensuremath{\sigma}}_{J+L,J+L+l}〉\ensuremath{\sim}〈{\ensuremath{\sigma}}_{00}{\ensuremath{\sigma}}_{\mathrm{NN}}〉〈{\ensuremath{\sigma}}_{00}{\ensuremath{\sigma}}_{\mathrm{LL}}〉+{〈{\ensuremath{\sigma}}_{00}{\ensuremath{\sigma}}_{\mathrm{NN}}〉}^{c}{〈{\ensuremath{\sigma}}_{00}{\ensuremath{\sigma}}_{\mathrm{LL}}〉}^{c}{f}_{4}({x}_{1},{x}_{2},X,\ensuremath{\varphi})〈\ensuremathσ00​\ensuremathσNN​\ensuremathσJ,J+l​\ensuremathσJ+L,J+L+l​〉\ensuremath∼〈\ensuremathσ00​\ensuremathσNN​〉〈\ensuremathσ00​\ensuremathσLL​〉+〈\ensuremathσ00​\ensuremathσNN​〉c〈\ensuremathσ00​\ensuremathσLL​〉cf4​(x1​,x2​,X,\ensuremathφ), where f4{f}_{4}f4​ is the scaling function. The scaling variables x1,x2{x}_{1},{x}_{2}x1​,x2​, and XXX are, respectively, the lengths of the pairs (NNN and LLL) and distance between the centers of the pairs scaled by the corresponding correlation lengths. The angle \ensuremathφ\ensuremath{\varphi}\ensuremathφ measures the deviation from the diagonal axis of the line joining the centers. The scaling function is given as a series of x1{x}_{1}x1​ and x2{x}_{2}x2​, truncated at the order of (x1x2)3{({x}_{1}{x}_{2})}^{3}(x1​x2​)3.
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We calculate the multispin correlation functions away from Tc{T}_{c}Tc​ of a two-dimensional Ising lattice along the diagonal. We also present the result for a four-spin correlation function in the scaling limit. These results show that with slight modifications, the operator reduction formulae of Kadanoff hold.
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Abstract In the neighbourhood of the critical point, the correlation length of the spin–spin correlation function of the two-dimensional Ising model diverges. The correlation function permits a scaling limit in which the separation N between spins goes to infinity, but the scaling variable remains fixed, where t is the coupling, and t = 1 the critical point. Previous work has specified these scaling functions (there is one for the critical point being approached from above, and another if approached from below) in terms of transcendents defined by a particular -form of the degenerate Painlevé V equation. For the diagonal–diagonal correlation, we characterise the first two leading large N correction terms to the scaling functions—these occur at orders N −1 and N −2 —in terms of solutions of a second order linear differential equation with coefficients given in terms of these transcendents, and show how they can be computed. We show that the order N −1 is trivial and can be eliminated through appropriate variables so that the leading non-trivial correction is of order N −2 . In this respect our solution is the first non-trivial correction to the scaling limit pair-correlation results given in the earlier literature, but restricted to the case with both spins on a diagonal.