Approximate critical surface of the bond-mixed square-lattice Ising model

Abstract

The critical surface of the quenched bond-mixed square-lattice spin-$\frac{1}{2}$ first-neighbor-interaction ferromagnetic Ising model (with exchange interactions ${J}_{1}$ and ${J}_{2}$) has been investigated. Through renormalization group and heuristical procedures, a very accurate [error inferior to 3 \ifmmode\times\else\texttimes\fi{} ${10}^{\ensuremath{-}4}$ in the variables ${t}_{i}\ensuremath{\equiv}tanh(\frac{{J}_{i}}{{k}_{B}T})$] approximate numerical proposal for all points of this surface is presented. This proposal simultaneously satisfies all the available exact results concerning the surface, namely ${p}_{c}=\frac{1}{2}$, ${t}_{c}=\sqrt{2}\ensuremath{-}1$, both limiting slopes in these points, and ${t}_{2}=\frac{(1\ensuremath{-}{t}_{1})}{(1+{t}_{1})}$ for $p=\frac{1}{2}$. Furthermore an analytic approximation [namely, $(1\ensuremath{-}p)\mathrm{ln}(1+{t}_{1})+p\mathrm{ln}(1+{t}_{2})=\frac{1}{2}\mathrm{ln}2$] is also proposed. In what concerns the available exact results, it only fails in reproducing one of the two limiting slopes, where there is an error of 1% in the derivative: These facts result in an estimated error less than ${10}^{\ensuremath{-}3}$ (in the $t$ variables) for any point in the surface.