Ising-Model Reformulation. III. Quadruplet Spin Averages

This is the authors' abstract. We don't add key points for this paper.

A previously developed method for diagrammatic expansion of the Ising-model partition function and pair distribution function is applied to calculation of the field-free quadruplet spin averages 〈μ₁μ₂μ₃μ₄〉. The central four-vertex function Q generated by these averages is topologically analyzed in standard fashion in terms of an irreducible four-vertex quantity e. Several rigorously necessary conditions that must be satisfied by Q are listed. It is furthermore pointed out that the existence of a logarithmic specific-heat anomaly puts an additional constraint on quadruplet averages (which is verified in an Appendix by direct calculation on the Ising-Onsager two-dimensional square lattice). Upon making a simplifying functional assumption for Q, this latter quantity may be entirely determined above Tc by one of the necessary conditions. This leads at Tc to a kd/2 spectrum (d=dimensionality) and above Tc to a logarithmic specific heat, both of which were adduced by Abe's Ising-model version of the He⁴ speculative analysis due to Patashinskii and Pokrovskii (but for different reasons from those in the present analysis). Sinec the Abe-Patashinskii-Pokrovskii spectrum almost certainly exhibits an incorrect exponent, an alternative and more powerful functional assumption for Q is suggested which still yields a soluble theory in principle, but construction of the solution is not attempted here.