One-Dimensional Ising Model with Random Exchange Energy

Abstract

The one-dimensional Ising model with random exchange energy is formulated in terms of a homogeneous integral equation. Assuming the input distribution has a narrow width proportional to $N^{-1\mathrm{}}$, the integral equation is solved by perturbation in $N^{-1\mathrm{}}$. The shift in the free energy of the system up to the order of $N^{-2\mathrm{}}$ is given. It is found that for a symmetrical distribution, the shift due to the randomness is second order in $N^{-1\mathrm{}}$ and negative, depending only upon the variance of the input distribution. The first-order shift for the asymmetric distribution comes entirely from the asymmetry. After a shift in energy is made to account for this asymmetry, the effective shift is identical to that for the symmetrical case. The shifts of all the thermodynamic properties of the system are also given. The randomness is found to decrease the magnetization for all temperature and applied field. However, shifts in magnetic susceptibility and specific heat are oscillatory in sign.