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Physical Review Letters

Correlation Functions in a Corner-Shaped Ising Model

Douglas B. Abraham, F. T. Latrémolière

Published Jun 17, 1996• 5 citations
Publisher

Abstract

We obtain exact results at all temperatures for the two-point function of surface spins near the corner of a two-dimensional rectangular Ising ferromagnet; at the critical point, the functional form predicted by conformal theory is recovered. We are also able to calculate exactly a large class of correlation functions involving spins off the surface, and to interpret the results geometrically.

DOI
10.1103/physrevlett.76.4813

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We obtain exact results at all temperatures for the two-point function of surface spins near the corner of a two-dimensional rectangular Ising ferromagnet; at the critical point, the functional form predicted by conformal theory is recovered. We are also able to calculate exactly a large class of correlation functions involving spins off the surface, and to interpret the results geometrically.
By solving partial differential equations for the correlation functions implied by conformal invariance, we obtain exact results at the critical point for the bulk six-point function 〈\ensuremath{\sigma}(1)...\ensuremath{\sigma}(6)〉 of the two-dimensional Ising model and the three-point functions 〈\ensuremath{\sigma}(1)\ensuremath{\sigma}(2)\ensuremath{\sigma}(3)〉, 〈\ensuremath{\sigma}(1)\ensuremath{\mu}(2)\ensuremath{\mu}(3)〉 of the semi-infinite two-dimensional Ising model with fixed boundary spins. Here \ensuremath{\sigma} and \ensuremath{\mu} denote order (spin) and disorder variables, respectively. We also derive the explicit form of the magnetization profile at the bulk critical temperature in an Ising strip of infinite length and finite width, with spin-up boundary conditions on one edge and spin-down boundary conditions on the other.
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 two-spin and four-spin correlation functions involving one and two pairs of nearest neighbours respectively are evaluated near the free surface of an Ising ferromagnet and the departure from the bulk value is found. These correlation functions could be termed the energy density and the energy density-energy density correlation functions. Near the critical temperature T c the energy density in the surface layer is found to behave as t 2 ln t with t= mod T-T c mod . This contrasts with the bulk energy density which has a t ln t singularity. Correlations between a surface and a bulk spin are evaluated above and below T c . Correlations between surface spins when the coupling between the surface spins is different from those in the bulk are examined.
We consider the anisotropic Ising model on the triangular lattice with finite boundaries, and use Kaufman's spinor method to calculate low-temperature series expansions for the partition function to high order. From these we can obtain 108-term series expansions for the bulk, surface and corner free energies. We extrapolate these to all terms and thereby conjecture the exact results for each. Our results agree with the exactly known bulk free energy. For the isotropic case, they also agree with Vernier and Jacobsen's conjecture for the 60∘60^{\circ}60∘ corners, and with Cardy and Peschel's conformal invariance predictions for the dominant behaviour at criticality.
We consider the fully frustrated Villain or ``odd'' Ising model on a semi-infinite square lattice with surface couplings JS{\mathit{J}}_{\mathit{S}}JS​ and bulk couplings \ifmmode\pm\else\textpm\fi{}J. The zero-temperature properties of this model have an interpretation in terms of dimers. From previous work it is known that at T=0 there is no bulk long-range order and the bulk spin-spin correlation function decays algebraically with critical exponent \ensuremath{\eta}=1/2. We analyze the surface behavior in the T=0 transition. For JS{\mathit{J}}_{\mathit{S}}JS​\ensuremath{\ne}1/2J the surface spins do order at T=0, and the connected pair-correlation function decays as ${\mathit{r}}^{\mathrm{\ensuremath{-}}2}$ parallel to the surface. For JS{\mathit{J}}_{\mathit{S}}JS​=1/2J there is no long-range surface order at T=0, and the pair-correlation function of the surface spins decays as ${\mathit{r}}^{\mathrm{\ensuremath{-}}1}$ and ${\mathit{r}}^{\mathrm{\ensuremath{-}}3}$ for r even and odd, respectively. We also calculate the free energy, the correlation length, and the profile of the energy density in strips of finite width and compare the results with predictions of conformal invariance for ordinary critical systems.