Numerical determination of the critical isotherm of the ferromagnetic Ising model

Abstract

The low-density linked-cluster expansion of the ferromagnetic Ising model F(H,T)=-mH-1/2qJ-k_BTΣl=1∞L_l(u)µ^l (u=e-^4J/kB^T,µ=e^-2mH/kB^T) is numerically analyzed. We find that all the polynomials L_l are positive between 0 and u_c=e^-4J/kB^Tc, T_c being the critical temperature, and that u_c can be calculated with good accuracy as the limit point of the zeros of these polynomials L_l. Given u_c, we compute L_l(u_c) 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,T_c)∼(-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.