n-spin correlation functions for the tvvo-dimensional Ising model*

It is noted that expressions for the T = T, . n-spin correlation functions of a two-dimensional Ising system obtained by continuum methods do agree with exact results for restricted geometries. This leads to the conjecture that the continuum result is true for all n and all geometries. Specula-tion regarding the behavior of these functions away from T = T, . are made.


I. INTROOU. CTION
Although the two-spin correlation function for the two-dimensional Ising model has been known for some time, " such a knowledge of n-spin correlation functions is not available.A formal expression for such functions away from the critical point has been obtained by McCoy, Tracy, and Wu.' It is not im- mediately obvious how to apply their results to T = T, At T = T, .exact results are known for restrictive geometries.Kadanoff and Ceva' and Pink4 found the n-point function for the collinear points along one of the lattice directions.This result has been extended by Au-Yang' to collinear points along a diagonal.For spin separations larger than the lattice constant the above results are Il (o(r~) a(r"))r=r =(const) g ( r, -r, (' " not agree.Equation (2) looks much more symmetric.
We wish to point out the embarrassingly simple fact that they do agree.For n =4 it is a matter of trivial algebra.The proof for n & 4 will be given in Sec.II.
Finally, we have compared Eq. (2} for the case n =4 to the results for a parallel set of points present- ed in Ref. 6.The two expressions agree to the order the results are stated in Ref. 6.These agreements lend strong support to the conjecture that Eq. ( 2) for the square of the correlation function is valid for all n and all geometric configurations.
An alternate approach has been to work directly in the continuum limit using the analogy of the Ising problem to a two dimensional fermion field theory.' " Luther and Peschels gave the result for an arbitrary set of n spins located at r, : (a(r, ) .a(r"))'r=r =(const) X ' g ) r, -r, (""' . (2) +. 1 IA j I The prime in the summation indicates that the configurations of the q, 's are restricted to the ones satisfying gg, =0.Note that Eq. ( 2) is for the square of the correlation function, The same expression was obtained in Ref. 9 for collinear points and using similar techniques for noncollinear points." At first glance it appears that Eqs.(I) and (2) do where the expectation of the right-hand side is taken in the vacuum of a sine-Gordon field theory with a Lagrangian L = -, ((ig}' -m/rr: cos2 Jvr p: (4) with m = -(T -T, .).The above may serve as a basis 1 for a mass perturbation away from T = T, I I. PROOF (cr(x)) rr(xt) II a. (x"))' = (const) g (x; -x, )' " '"   We wish to show that Eq. ( 2) and the square of Eq.
(1) are the same for the geometric situation ~here all the r, 's are collinear.As noted above, the observation for n =4 requires the trivial algebra of bringing Eq.
(2) to a common denominator.For n & 4, " the argu- ment we found is somewhat baroque.
We study the fourth power of the correlation func- tion.From Eq. {1)we find n-SPIN CORRELATION FUNCTIONS FOR THE. . .
The square of Eq. ( 2) is likewise a sum of such terms.
The proof of the equality of the two expressions proceeds by induction.%'e note that both Eqs. ( 1) and (2) are invariant under separate permutations of where the x, 's are placed in increasing order.This is, ho~ever, the expression for the vacuum expectation value of a product of free fermion fields" and thus can be expressed as a sum of products of two-body propagators: {a. (x, }a (x,} a (x") }' =l(oil{xi)0 (x&) p{x&) pt(x") lol the x, 's with odd or even i' s.At this stage we use fac- torization.Take any product of fermion propagators that appears in the decomposition of the square of Eq.
(2).First, we would like to rule out terms with propa- gators 1/{x, -x,), with both i and j simultaneously even, or odd, as such terms do not occur in Eq. ( 6).
Should such a term occur we could permute it to in- volve x, 's for i ~4.If we now separate the first four x's from the rest, factorization would imply that such a term would by present for correlation functions in- volving 4 or n -4 spins.Similarly the presence of the allowed terms can be checked using factorization and induction.
%'e wish to thank Dr. A. Luther for many valuable d&scussions.
Supported in part by the NSF.