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Journal of Physics A Mathematical and General

Correlation near the free surface of an Ising ferromagnet

P Reed

Published Jan 1, 1978• 11 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/11/1/015

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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.
The authors present a simple argument showing that the surface energy density of a semi-infinite d-dimensional spin system has, in general, a leading thermal singularity of the same form, mod T-T c mod 2- alpha , as the bulk free energy. At bulk criticality energy-energy correlations decay as r -2d parallel to the surface. These results hold for both free and fixed boundary spins, i.e. the 'ordinary' and 'extraordinary' transitions. The extraordinary critical behaviour of the magnetisation is the same as that of the energy density. They also confirm and generalise these results in two dimensions with an independent approach based on conformal invariance.
We consider a semi-infinite two-dimensional Ising model with nearest-neighbor coupling constants that deviate from the bulk coupling by $A{m}^{\ensuremath{-}y}$ for large mmm, mmm being the distance from the edge. The case A<0A<0A<0 of couplings which are weaker near the surface has been discussed by Hilhorst and van Leeuwen. We report exact results for the boundary magnetization and boundary pair-correlation function when A>0A>0A>0. At the bulk critical temperature there is a rich variety of critical behavior in the $A \ensuremath{-}y$ plane with both paramagnetic and ferromagnetic surface phases. Some of our results can be derived and generalized with simple scaling arguments.
Results are derived for the dynamic correlation functions between a pair of spin operators at any two sites within a semi-infinite Heisenberg ferromagnet at low temperatures, T<<Tc. The correlation functions are found to contain terms describing the effects of the bulk spin-wave modes and the surface spin-wave modes, together with appropriate weighting factors. The results are employed to evaluate the spin deviation in the ferromagnet as a function of temperature and distance from the surface.
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.