Numerical determination of the critical isotherm of the ferromagnetic Ising model

The low-density linked-cluster expansion of the ferromagnetic Ising model F(H,T)=-mH-1/2qJ-kBTΣ∞l=1Lₗ(u)μˡ (u=e^-4JkBT,μ=e^-2mHkBT) is numerically analyzed. We find that all the polynomials Lₗ are positive between 0 and uc=e^-4JkBTc, Tc being the critical temperature, and that uc can be calculated with good accuracy as the limit point of the zeros of these polynomials Lₗ. Given uc, we compute Lₗ(uc) and find, numerically, their asymptotic behavior to be c₀l^-s. It is then shown that the singular part of the free energy per spin, as H→0, is given by F(H,Tc)~(-lnμ)^s-1~H^s-1. s is found to be about 2.061 in two dimensions and about 2.19 in three dimensions. Hence, the critical M-H isotherm is found and the critical index δ is calculated in fair agreement with the previous estimates. An analytic expression for the corresponding critical amplitude is also found. The form of F given above is based on numerical work and is not to be regarded as proven analytically.