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arXiv:2005.02240 · Journal of Physics Conference Series

Phase Transitions in the Two-Dimensional Ising Model from the Microcanonical Perspective

Kedkanok Sitarachu, Michael Bachmann

Published Feb 1, 2020• 10 citations
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Abstract

Abstract The continuous ferromagnetic-paramagnetic phase transition in the two-dimensional Ising model has already been excessively studied by conventional canonical statistical analysis in the past. We use the recently developed generalized microcanonical inflection-point analysis method to investigate the least-sensitive inflection points of the microcanonical entropy and its derivatives to identify transition signals. Surprisingly, this method reveals that there are potentially two additional transitions for the Ising system besides the critical transition.

DOI
10.1088/1742-6596/1483/1/012009

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Abstract The continuous ferromagnetic-paramagnetic phase transition in the two-dimensional Ising model has already been excessively studied by conventional canonical statistical analysis in the past. We use the recently developed generalized microcanonical inflection-point analysis method to investigate the least-sensitive inflection points of the microcanonical entropy and its derivatives to identify transition signals. Surprisingly, this method reveals that there are potentially two additional transitions for the Ising system besides the critical transition.
Abstract Recent analyses of least-sensitive inflection points in derivatives of the microcanonical entropy for the two-dimensional Ising model revealed higher-order transition signals in addition to the well-studied second-order ferromagnetic/paramagnetic phase transition. In this paper, we re-analyze the exact density of states for the one-dimensional Ising chain as well as the strips with widths/lengths of up to 64/1024 spins, in search of potential transition features. While some transitions begin to emerge as the strip width increases, none are found for the chain, as might be expected.
We employ the microcanonical inflection-point analysis method, developed for the systematic identification and classification of phase transitions in systems of any size, to study the two-dimensional Ising model at various lattice sizes and in the thermodynamic limit. Exact results for the density of states, which were obtained by exact algorithmic computation, provide evidence for higher-order transitions in addition to the well-studied second-order ferromagnetic-paramagnetic phase transition. An independent third-order phase transition is identified in the ferromagnetic phase, whereas another third-order transition resides in the paramagnetic phase. The latter is a dependent transition, i.e., it is inevitably associated with the critical transition, but it remains separate from the critical point in the thermodynamic limit. For a deeper insight into the nature of these additional transitions, a detailed analysis of spin clusters is performed.
Abstract This article offers a detailed analysis of pseudo-phase transitions of Ising and Baxter–Wu models in two-dimensional finite-size lattices. We carry out Wang–Landau sampling to obtain the density of states. Using microcanonical inflection-point analysis with microcanonical entropy, we obtain the order of the pseudo-phase transitions in the models. The microcanonical analysis results of the second-order transition for the Ising model and the first-order transition for the Baxter–Wu model are consistent with the traditional canonical results. In addition, the third-order transitions are found in both models, implying the universality of higher-order phase transitions.
Traditional definition of phase transition involves an infinitely large system in thermodynamic limit. Finite systems such as biological proteins exhibit cooperative behavior similar to phase transitions. We employ recently discovered analysis of inflection points of microcanonical entropy to estimate the transition temperature of the phase transition in q state Potts model on a finite two dimensional square lattice for q=3 (second order) and q=8 (first order). The difference of energy density of states (DOS) Δ ln g(E) = ln g(E+ ΔE) −ln g(E) exhibits a point of inflexion at a value corresponding to inverse transition temperature. This feature is common to systems exhibiting both first as well as second order transitions. While the difference of DOS registers a monotonic variation around the point of inflexion for systems exhibiting second order transition, it has an S-shape with a minimum and maximum around the point of inflexion for the case of first order transition.
The basic quantity for the description of the statistical properties of physical systems is the density of states or, equivalently, the microcanonical entropy. Macroscopic quantities of a system in equilibrium can be computed directly from the entropy. Response functions such as the susceptibility are related, for example, to the curvature of the entropy surface. Interestingly, physical quantities in the microcanonical ensemble show characteristic properties of phase transitions in finite systems. In this article we investigate these characteristics for finite Ising systems. The singularities in microcanonical quantities, which signal a continuous phase transition in infinite systems, are characterized by classical critical exponents. Estimates of the non-classical exponents which emerge only in the thermodynamic limit can nevertheless be obtained by analysing effective exponents or by applying a microcanonical finite-size scaling theory. This is explicitly demonstrated for two- and three-dimensional Ising systems.
This article investigates the pseudo transitions of the Blume-Capel model on two-dimensional finite-size lattices. By employing the Wang-Landau sampling method and microcanonical inflection point analysis, we identified the positions of phase transitions as well as higher-order phase transitions. Through Metropolis sampling and canonical ensemble analysis, we determined the geometric characteristics of the system at these transition points. When the crystal field parameter DDD exceeds 1.965, crossing the tricritical point, no third-order dependent phase transition is observed. However, a fourth-order independent transition was identified in the high-temperature region, and through Metropolis sampling analysis, a phase transition from the ordered paramagnetic phase to the disordered paramagnetic phase was confirmed, enhancing the phase diagram. Furthermore, the positions of the third-order phase transition obtained from both microcanonical and canonical analyses are consistent and mutually corroborative. We speculate that third-order dependent transitions vanish in the presence of strong first-order phase transitions.