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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 Oct 7, 2022• 1 citation
Publisher

Abstract

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.

DOI
10.1142/s2010324722500266

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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.
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 approach to determination of the order parameter in the finite 1D Ising model was proposed. For such systems a root mean square magnetization and an average size of the maximal ground state cluster which is formed by neighboring spins with an energy {-2J} does not disappear at T!=0. The temperature dependence of heat capacity and its derivative with respect to temperature from the average size of the maximum quasi ground state cluster, formed by the neighboring spins with an energy {-2J,0} in the 1D Ising chain were considered. The percolation approach in Curie-Weiss does not work due the internal structure of the model.
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.
Determining the phase diagram of interacting quantum many-body systems is an important task for a wide range of problems such as the understanding and design of quantum materials. For classical equilibrium systems, the Lee-Yang formalism provides a rigorous foundation of phase transitions, and these ideas have also been extended to the quantum realm. Here, we develop a Lee-Yang theory of quantum phase transitions that can include thermal fluctuations caused by a finite temperature, and it thereby provides a link between the classical Lee-Yang formalism and recent theories of phase transitions at zero temperature. Our methodology exploits analytic properties of the moment generating function of the order parameter in systems of finite size, and it can be implemented in combination with tensor-network calculations. Specifically, the onset of a symmetry-broken phase is signaled by the zeros of the moment generating function approaching the origin in the complex plane of a counting field that couples to the order parameter. Moreover, the zeros can be obtained by measuring or calculating the high cumulants of the order parameter. We determine the phase diagram of the two-dimensional quantum Ising model and thereby demonstrate the potential of our method to predict the critical behavior of two-dimensional quantum systems at finite temperatures.
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.