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arXiv:1911.05475

Unveiling phase transitions in 1D systems with short-range interactions

L. S. Ferreira, L. N. Jorge, Cláudio José DaSilva, Minos A. Neto, Álvaro A. Caparica

Published Nov 13, 2019• 0 citations
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Abstract

The statement that any phase transition is related to the appearance or disappearance of long-range spatial correlations precludes a finite transition temperature in one-dimensional (1D) systems. In this paper we demonstrate that the 1D Ising model with short-range exchange interactions exhibits a second-order phase transition at a finite temperature relying on the proper choice of the order parameter. To accomplish this, we combined analytical calculations and high-precision entropic sampling simulations and chose a slightly different order parameter, namely the module of the magnetization. Notably, we detected a phase transition with a corresponding critical temperature around 15 K, which is in excellent agreement with experimental results. Our study indicates that an inappropriate choice of the order parameter may mask phase transitions in one-dimensional systems.

DOI
10.48550/arxiv.1911.05475

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The statement that any phase transition is related to the appearance or disappearance of long-range spatial correlations precludes a finite transition temperature in one-dimensional (1D) systems. In this paper we demonstrate that the 1D Ising model with short-range exchange interactions exhibits a second-order phase transition at a finite temperature relying on the proper choice of the order parameter. To accomplish this, we combined analytical calculations and high-precision entropic sampling simulations and chose a slightly different order parameter, namely the module of the magnetization. Notably, we detected a phase transition with a corresponding critical temperature around 15 K, which is in excellent agreement with experimental results. Our study indicates that an inappropriate choice of the order parameter may mask phase transitions in one-dimensional systems.
Abstract The statement that any phase transition is related to the appearance or disappearance of long-range spatial correlations precludes a finite transition temperature in one-dimensional (1D) systems. In this paper we demonstrate that the 1D Ising model with short-range exchange interactions exhibits a second-order phase transition at a finite temperature relying on the proper choice of the order parameter. To accomplish this, we combined analytical calculations and high-precision entropic sampling simulations and chose a slightly different order parameter, namely the module of the magnetization. Notably, we detected a phase transition with a corresponding critical temperature around 15 K, which is in excellent agreement with experimental results. Our study indicates that an inappropriate choice of the order parameter may mask phase transitions in one-dimensional systems.
The statement that any phase transition is related to the appearance or disappearance of long-range spatial correlations and precludes a finite transition temperature in one-dimensional (1D) systems is rooted in condensed matter theory. In this paper, we demonstrate that the 1D Ising model with short-range exchange interactions exhibits a second-order phase transition at a finite temperature relying on the proper choice of the order parameter. We combined analytical calculations and high-precision entropic sampling simulations and chose a slightly different order parameter, namely, the modulus of the magnetization. Notably, we detected a phase transition with a corresponding critical temperature around 15 K, which is in excellent agreement with experimental results. Our study indicates that the incorrect choice of the order parameter may mask phase transitions in 1D systems.
Ordering of a one-dimensional stochastic Ising model evolving by a combination of spin flips and spin exchanges is investigated. The spin-flip rates satisfy detailed balance for the equilibrium state of the Ising model at temperature T, while the spin exchanges are random and of arbitrary range. Analytical methods and Monte Carlo simulations are used to show that, depending on the details of the spin-flip rate, finite-temperature phase transition may or may not occur in the system. When ordering occurs, it is of mean-field type and the scaling function describing the finite-size scaling of the magnetization fluctuations is found to be indistinguishable from that of an equilibrium Ising model with infinite-range interaction.
The critical behavior of the Ising model on a one-dimensional network, which has long-range connections at distances l>1 with the probability theta(l) approximately l(-m), is studied by using Monte Carlo simulations. Through analyzing the Ising model on networks with different m values, this paper discusses the impact of the global correlation, which decays with the increase of m, on the phase transition of the Ising model. Adding the analysis of the finite-size scaling of the order parameter [M], it is observed that in the whole range of 0<m<2, a finite-temperature transition exists, and the critical exponents show consistence with mean-field values, which indicates a mean-field nature of the phase transition.
As is well known, one-dimensional spin systems with short-range interactions cannot undergo a phase transition at finite temperature. However, in many 1D Ising spin models with frustration, at a certain crossover point, the thermodynamic quantities show anomalies that resemble such a transition W. Yin [Phys. Rev. Res. 6, 013331 (2024)2643-156410.1103/PhysRevResearch.6.013331]. Recently, Yin [Phys. Rev. Res. 6, 013331 (2024)2643-156410.1103/PhysRevResearch.6.013331] has shown, using decorated ladders as an example, that a one-dimensional Ising model can be arbitrarily close to the genuine phase transition. Using the transfer matrix method and real space renormalization group technique, we find that crossover resembling a phase transition accompanied by gigantic specific heat can also occur in two-dimensional systems. We consider coupled Ising spin chains with a zig-zag structure and a four-spin j_{4} interaction, which allows us to control the temperature at which the spins become unbound in the model with short-range interactions. The crossover point is defined by two parameters: the crossover temperature t=t^{*} and the crossover interaction j_{4}=j_{4}^{*}, and occurs when the temperature at which the effective interaction between the spin subsystems zeros out coincides with the temperature at which the spins become unbound. In systems in which a genuine phase transition occurs at the critical temperature t=t_{c}, there are two possibilities: if t_{c}>t^{*} the system undergoes a sequence of three phase transitions with a reentrant transition to disordered phase, while for t_{c}<t^{*} the phase transition is preceded by the considered crossover. We study the key role in the appearance of anomalies in the thermodynamic quantities of the four-spin interactions which are suspected to have a significant impact on the properties of magnetic two-dimensional lattices, but possible also on the construction of social systems modeling the phenomenon of social validation or peer pressure.
The critical behaviour of the one-dimensional Ising model with long-range ferromagnetic interactions decaying with distance r as 1/r 1+ sigma has been studied by scaling the range of interactions. Exact calculations have been done for a system with finite ranges up to the tenth neighbour for 0< sigma <or=1. Applying the range scaling, the critical temperature, critical exponent nu and the anomalous dimension of the order parameter have been calculated. Additional analysis of the convergence of the method has been performed by applying the Vanden Broeck and Schwartz extrapolation procedure in addition to the simple least-squares approximation and by evaluating the convergence exponents.