ON THE CRITICAL POLYNOMIAL OF THE SIMPLE CUBIC ISING-MODEL

In 1985 Rosengren conjectured that the critical point of the symmetric , simple cubic (SC) Ising model is given by upsilon(c) = tanh(J/k(B)T( c)) = upsilon(R) = (root 5 - 2) cos(pi/8). This guess is examined in t he context of attempting to construct the full critical polynomial P-3 (upsilon(x), upsilon(y), upsilon(z)), with a root upsilon(c)(J(x), J(y ), J(z)), for the anisotropic SC Ising model with couplings J(x), J(y) and J(z). It transpires that upsilon(R) is a surd which satisfies R(u psilon(R)(2)) = 0, where R(x) is a quartic polynomial with integral co efficients; but R(upsilon(2)) is a poor candidate for P-3(upsilon, ups ilon, upsilon) since it does not display various 'nice' properties emb odied in the critical polynomial P-2(upsilon(x), upsilon(y)) for the s quare, 2D Ising lattices. Methods for constructing nice polynomials Q( k)(upsilon(x), upsilon(y), upsilon(z)) that provide excellent approxim ations for upsilon(c) and for upsilon(R) are demonstrated. However, sc aling arguments, etc, for the dimensional crossover induced when, say, J(z) --> 0 cast doubt on the existence and nature of the sought-for c ritical polynomial P-3.