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arXiv:1809.09240 · Journal of Statistical Mechanics Theory and Experiment

Leading corrections to the scaling function on the diagonal for the two-dimensional Ising model

Peter J. Forrester, Jacques H. H. Perk, Allan K. Trinh, Nicholas S. Witte

Published Feb 1, 2019• 7 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1742-5468/ab00e6

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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.
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We study the asymptotic behavior, for large separations, of the spin-spin correlation function $〈{\ensuremath{\sigma}}_{0,0}{\ensuremath{\sigma}}_{M,N}〉$ in the two-dimensional Ising model, where the two spins are not necessarily on the same row. Besides the limiting value for infinite separation, which is the square of the spontaneous magnetization, we evaluate the two leading terms in the asymptotic expression in each of the two cases T<TcT<{T}_{c}T<Tc​ and T>TcT>{T}_{c}T>Tc​. It is found that the nearest singularity of the generating function for the correlation is quite simple in the case T>TcT>{T}_{c}T>Tc​, but much more complicated for T<TcT<{T}_{c}T<Tc​. In an Appendix, we also give exactly in a very simple form the correlation $〈{\ensuremath{\sigma}}_{0,0}{\ensuremath{\sigma}}_{N,N}〉$ for symmetrical Ising lattice at the critical temperature Tc{T}_{c}Tc​.
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 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 $\ensuremath{\varphi}$ 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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