Ising-Model Reformulation. III. Quadruplet Spin Averages

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 $〈{\mu }_1{\mu }_2{\mu }_3{\mu }_4〉$. 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 $T_c$ by one of the necessary conditions. This leads at $T_c$ to a $k^{\frac{d}{2}}$ spectrum ($d=\mathrm{dimensionality}$) and above $T_c$ to a logarithmic specific heat, both of which were adduced by Abe's Ising-model version of the ${\mathrm{He}}^4$ 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.