Critical behaviour of Ising systems on extremely anisotropic lattices: V. Some aspects of the cross-over from d to d - 1 dimensions

Abstract

The critical equation of a ferromagnetic spin- 1 2 Ising system on a d-dimensional cartesian lattice with coupling constants J 1, …, J d along the d lattice axes is investigated by means of a series expansion for the reduced initial susceptibility χ' d in the variable t d = tanh βJ d . The coefficients a n in this series are sums of products of multiple-spin correlation functions on a ( d − 1)-dimensional cartesian lattice with coupling constants J 1, …, J d − 1 . It is shown that a 0 = χ' d − 1 , a 1 = 2 χ' 2 d − 1 , 2 χ' 3 d − 1 ≤ 4 χ' 3 d − 1 , 0 ≤ a 3 ≤ 8 χ' 4 d − 1 , where χ' d − 1 is the susceptibility of the Ising system on the ( d − 1)-dimensional lattice; for the critical exponent γ ( n) of a n ( n ≤ 3) this implies: γ (0) = γ d − 1 , γ (1) = 2 γ d − 1 , γ (2) = 3 γ d − 1 , γ (3) ≤ 4 γ d − 1 , where γ d − 1 is the critical exponent of χ' d − 1 . On the basis of the conjecture that γ ( n) = ( n + 1) γ d − 1 and a symmetry argument the equation (1 − 2x 2) 7 4 + (1 − 2x 3) 7 4 = 1, [x 2(3) = tanh βJ 2(3)/(1 − tanh βJ 1)] , is proposed for the asymptotic form of the critical equation of the simple cubic lattice in the limit J 2/ J 1 → 0, J 3/ J 1 → 0; this equation is in full agreement with numerical results obtained in a previous paper (II).