Numerical determination of the critical isotherm of the ferromagnetic Ising model

The low-density linked-cluster expansion of the ferromagnetic Ising model $F(H,T)=-mH-\frac{1}{2}qJ-k_{B}T\Sigma \stackrel{\infty }{l=1\mathrm{}}{L}_l\mathrm{}\left(u\right)\mathrm{}{\mu }^l$ ($u=e^{\frac{-4J}{k_{B}T}},\mu =e^{\frac{-2\mathrm{mH}}{k_B}T}$) is numerically analyzed. We find that all the polynomials $L_l$ are positive between 0 and $u_c=e^{\frac{-4J}{k_B{T}_c}}$, $T_c$ 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 $L_l\left(u_c\right)$ and find, numerically, their asymptotic behavior to be $c_0{l}^{-s}$. It is then shown that the singular part of the free energy per spin, as $H\to 0$, is given by $F(H,T_c)\sim {(-\ln \mu )}^{s-1\mathrm{}}\sim H^{s-1\mathrm{}}$. $s$ 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 $\delta$ 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.