Critical temperature of one-dimensional Ising model with long-range interaction revisited

Abstract

We present a generalized expression for the transfer matrix of finite and infinite one-dimensional spin chains within a magnetic field with spin pair interaction J/rp, where r∈{1,2,3,…,nv} is the distance between two spins, nv is the number of nearest neighbors reached by the interaction, and 1≤p≤2. Using this transfer matrix, we calculate the partition function, the Helmholtz free energy, and the specific heat for both finite and infinite ferromagnetic 1D Ising models within a zero external magnetic field. We focus on the temperature Tmax where the specific heat reaches its maximum, needed to compute J/(kBTmax) numerically for every value of nv∈{1,2,3,…,30}, which we interpolate and then extrapolate up to the critical temperature as a function of p, by using a novel inter-extrapolation function. We use two different procedures to reach the infinite spin chain with an infinite interaction range: increasing the chain size as well as the interaction range by the same amount and increasing the interaction range for the infinite chain. As we expected, both extrapolations lead to the same critical temperature within their uncertainties by two different concurrent curves. Critical temperatures fall within the upper and lower bounds reported in the literature, showing a better coincidence with many existing approximations for p close to 1 than the p values near 2. It is worth mentioning that the well-known cases for nearest (original Ising model) and next-nearest neighbor interactions are recovered doing nv=1 and nv=2, respectively.