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

TL;DR

This paper derives a generalized transfer matrix for 1D Ising models with long-range interactions, computes critical temperatures, and compares numerical extrapolations to existing bounds, improving understanding of phase transitions in such systems.

Contribution

It introduces a novel generalized transfer matrix approach for long-range 1D Ising models and provides new numerical estimates of critical temperatures as a function of interaction range and decay exponent.

Findings

Critical temperatures fall within known bounds.

Better agreement with existing approximations near p=1.

Recovered known cases for near and next-near neighbor interactions.

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 $J/r^p$, where $r\ = 1,2,\ldots,n_v$ is the distance between two spins, $n_v$ is the number of nearest neighbors reached by the interaction, and $p \in [1,2]$. 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 $T_{\text{max}}$ where specific heat reaches its maximum. We calculate $J/(k_B T_{\text{max}})$ numerically for every values of $n_v \in \{ 1,2,\ldots, 25\}$, 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. As we expected, both extrapolations lead to the same critical temperature, although by two different concurrent curves. 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. We report an averaged inverse critical temperature $J/(k_BT_c) = 0.532$ for the one-dimensional spin chain with $p=2$. It is worth mentioning that the well-known cases for near (original Ising model) and next-near neighbor interactions are recovered doing $n_v = 1$ and $n_v = 2$, respectively.