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

The critical surface of the quenched bond-mixed square-lattice spin-1/2 first-neighbor-interaction ferromagnetic Ising model (with exchange interactions J₁ and J₂) has been investigated. Through renormalization group and heuristical procedures, a very accurate [error inferior to 3 ×{} 10^-4 in the variables tᵢ≡tanh(JᵢkBT)] approximate numerical proposal for all points of this surface is presented. This proposal simultaneously satisfies all the available exact results concerning the surface, namely pc=1/2, tc=√2-1, both limiting slopes in these points, and t₂=(1-t₁)(1+t₁) for p=1/2. Furthermore an analytic approximation [namely, (1-p)ln(1+t₁)+pln(1+t₂)=1/2ln2] 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^-3 (in the t variables) for any point in the surface.