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

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/r^p, where r = 1,2,...,n_v is the distance between two spins, n_v is the number of nearest neighbors reached by the interaction. With this generalized expression, 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 T_max where specific heat reaches its maximum. We calculate J/(k_B T_max) numerically for every values of n_v in {1,2,..., 25}, which we interpolate and then extrapolate up to the critical temperature as a function of p, using a novel functional approach. Two different procedures are used to reach the infinite spin chain with an infinite interaction range: increasing the chain size and the interaction range by the same amount, and increasing the interaction range for the infinite chain. Our critical temperatures as a function of p fall within the upper and lower bounds reported in the literature and show a better coincidence with many existing approximations for p close to 1 than for the p values near 2.