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

Highlights

• Critical temperatures of one-dimensional Ising model with long-range interactions obtained by transfer matrix method.

• From the specific heat maxima to the critical temperature in a one-dimensional Ising model with long-range interactions.

• Finite range interaction and chain finite-size effects are strongly dependent on the power-law interaction exponent.

• Different types of phase transitions associated with different power-law interaction exponents.

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 $r\in \{1,2,3,\dots ,n_v\}$ is the distance between two spins, $n_v$ is the number of nearest neighbors reached by the interaction, and $1\le p\le 2$. 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 $T_{\text{max}}$ where the specific heat reaches its maximum, needed to compute $J/\left(k_B{T}_{\text{max}}\right)$ numerically for every value of $n_v\in \{1,2,3,\dots ,30\}$, 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 $n_v=1$ and $n_v=2$, respectively.

Introduction

Since its solution in 1925, the Ising model has been studied in many facets to understand systems with interactions, paying special attention to phase transitions. Although the original 1D Ising model does not present a spontaneous magnetization at non-zero temperature, it has been proved [1], [2] that the Ising model with long-range interactions shows a phase transition at finite temperature. The first demonstration of a phase transition existence in the ferromagnetic long-range interaction Ising chain with an exchange interaction $\mathcal{J}\left(r\right)=J/r^p$ was given by Dyson [1] for $1<p<2$, as well as the non existence of phase transition for $p>2$ [3], [4] but keeping the case $p=2$ in doubts. The distance $r$ between interacting spins is given in units of the nearest neighbors and $J$ is a constant energy to which $\mathcal{J}\left(r\right)$ reduces when $p\to 0$. After some years, in 1982, Fröhlich and Spencer [2] proved the existence of a phase transition for $p=2$. Since the original demonstration of phase transition existence by Dyson, who did not provide a critical temperature, many approximations have been reported to give the critical temperatures for $1\le p\le 2$ using different methods: variational [5], coherent anomaly [6], $\zeta$-function [7], finite-range [8], series expansion [9], re-normalization group [10], [11], Monte Carlo [12], Bethe Lattice approximation [13], cluster approach [14], among others [15], [16], [17], [18], [19], [20], [21].

Moreover, the value of $p$ determines the type of transition on the system [20]. For $p\in (1,1.5)$, the system presents a phase transition of the mean-field type, i.e., the specific heat presents a jump at the transition [19], and the critical exponents are temperature independent as obtained by a mean-field approach [22]. For $p\in (1.5,2)$, the phase transition is non-mean-field type, and the specific heat presents a sharp peak [19]. For $p=2$, the system exhibits a jump in the magnetization [4], [23], which is known as the Thouless effect [24] and the spin–spin correlation function presents a power-law decay with a temperature-dependent exponent, which is related to the Kosterlitz–Thouless transition [25].

Although a phase transition can be identified by a discontinuity or divergence in the specific heat or the magnetic susceptibility, we focus on the specific heat since the magnetic susceptibility for spins systems has a very abrupt change at $h=0$, making it difficult its calculate. In addition, specific heat is a recurrently measured quantity in both physical and computational experiments to study phase transitions [26], [27], [28], [29], [30].

Despite a large amount of work to obtain the critical temperature of the long-range ferromagnetic 1D Ising model, there is still no exact analytical expression for its critical temperature. Our goal in this work is to give two new extrapolated critical temperature curves as functions of $p$ from the extrapolation of the temperatures at which the specific heat has a maximum. For this, we propose a novel inter-extrapolation function inspired by the Hurwitz zeta function. Inter-extrapolations are carried out in two ways: (i) from an infinite system with finite range interactions to a system of infinite range interactions, and (ii) from a finite system increasing the size and the range of interactions.

The rest of the article is organized as follows: In Section 2, we generalize the Kassan-Ogly transfer matrix [31] for the ferromagnetic long-range Ising model to include any number $n_v$ of nearest neighbors, which we use to obtain the corresponding partition function, the Helmholtz free energy and the specific heat expression used to calculate the temperature where the specific heat shows a maximum, which we called $T_{\text{max}}$. In Section 3, we use the set of $T_{\text{max}}$ obtained for each $n_v\in \{1,2,3,\dots ,30\}$ value to calculate the critical temperature as an extrapolation of $J/\left(k_B{T}_{\text{max}}\right)$ vs. $1/n_v$ as $n_v\to \infty$ for each $p\in \{1,1.05,1.1,1.2,1.3,\dots ,2\}$. In Section 4, we give our conclusions. We summarize our numerical calculations in Tables 1, 2, 3, and 4 given in Appendix A, and we give additional details of the calculation code and the comparisons of our inter-extrapolation functions in Appendix B.