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Theoretical and Mathematical Physics

Higher correlation functions of the Ising ferromagnet. The case of a linear model for spin S=1/2

M.P. Zhelifonov

Published Sep 1, 1971• 19 citations
Publisher
DOI
10.1007/bf01029349

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Progress of Theoretical Physics•Jun 1, 1966•15 citations
The theorem of Lee and Yang has been extended to the ferromagnetic Ising model with arbitrarily mixed spin values of Sj = ½, 1, and 32, including the case of equal spin values as a special one. Namely, it has been proved that the zeros of the partition function for the above Ising model with higher spin values lie on the unit circle in the fugacity plane (or complex magnetic-field plane). Expressions for general correlation functions in Ising ferromagnets with higher spin values have been derived in terms of the above generalized theorem. By the use of these expressions, the relations among the critical indices are discussed and the same results are obtained as those predicted by the scaling-law approach.
Using conformal field theory techniques, we compute the disorder-averaged p -th power of the spin-spin correlation function (overline⟨σ(0)σ( R )⟩ p , p ∊ Z ) for the ferromagnetic random bond Potts model. We thus generalize the calculations of Dotsenko, Dotsenko and Picco ( hep-th/9709136 , 1997), where the case p = 2 was considered, and of Ludwig ( Nucl. Phys. B , 85 (1987) 97, 330 (1990) 639), where first-order computations where made for general p . Perturbative calculations are made up to second order in ( being proportional to the central charge deviation of the pure model from the Ising model value). The explicit dependence of the correlation function on p gives an upper bound for the validity of the -expansion, which seems to be valid, in the three-states case, only if p ⩽ 4.
The Green function method is applied to the Ising model of ferromagnetism. It is essential in this paper that we have solved the equations for the Green functions in terms of the correlation functions without using the approximation of decoupling the Green functions as is usually done. Although we get several rigorous relations between the many-spin correlation functions, we do not succeed in solving for the correlation functions, except in the case of a linear chain. We introduce the approximation of eliminating some of the correlation functions using the high temperature expansion, so as to reduce the number of the independent correlation functions. The Curie points of a square lattice and a simple cubic lattice are calculated.