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Journal of Statistical Mechanics Theory and Experiment

Monte Carlo study for the thermodynamic and dynamic phase transitions in the spin-S Ising model on Sierpiński carpet

Hoseung Jang, Mouhcine Azhari, Unjong Yu

Published Jan 1, 2024• 4 citations
Publisher

Abstract

Abstract We study the thermodynamic and dynamic phase transitions (TPT and DPT) of the spin- 1 / 2 and spin-1 Ising models on three graphs constructed on the Sierpiński carpet. This study employs Monte Carlo methods, specifically the Wolff and Metropolis algorithms, in conjunction with finite-size scaling analysis. By calculating the critical temperature and critical exponent ratio γ / ν associated with the TPT, we demonstrate that the three graphs exhibit an identical critical exponent ratio for both the spin- 1 / 2 and spin-1 Ising models within statistical error. Furthermore, we explore the kinetic Ising model by varying the period of the oscillating external magnetic field and verify the existence of the DPT. We find that the critical exponent ratio γ / ν for the DPT matches that of the TPT. Our results suggest that the critical exponents or their ratios for continuous phase transitions in equilibrium and non-equilibrium systems with short-range interactions are independent of the graph structure and interaction type, as long as the background space is the same.

DOI
10.1088/1742-5468/ad0a91

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Abstract We study the thermodynamic and dynamic phase transitions (TPT and DPT) of the spin- 1 / 2 and spin-1 Ising models on three graphs constructed on the Sierpiński carpet. This study employs Monte Carlo methods, specifically the Wolff and Metropolis algorithms, in conjunction with finite-size scaling analysis. By calculating the critical temperature and critical exponent ratio γ / ν associated with the TPT, we demonstrate that the three graphs exhibit an identical critical exponent ratio for both the spin- 1 / 2 and spin-1 Ising models within statistical error. Furthermore, we explore the kinetic Ising model by varying the period of the oscillating external magnetic field and verify the existence of the DPT. We find that the critical exponent ratio γ / ν for the DPT matches that of the TPT. Our results suggest that the critical exponents or their ratios for continuous phase transitions in equilibrium and non-equilibrium systems with short-range interactions are independent of the graph structure and interaction type, as long as the background space is the same.
The short-time critical relaxation of an Ising model on a Sierpinski carpet is investigated using Monte Carlo simulation. We find that when the system is quenched from high temperature to the critical temperature, the evolution of the order parameter and its persistence probability, the susceptibility, and the autocorrelation function all show power-law scaling behavior at the short-time regime. The results suggest that the spatial heterogeneity and the fractal nature of the underlying structure do not influence the scaling behavior of the short-time critical dynamics. The critical temperature, dynamic exponent z, and other equilibrium critical exponents beta and nu of the fractal spin system are determined accurately using conventional Monte Carlo simulation algorithms. The mechanism for short-time dynamic scaling is discussed.
We study the critical behavior of the transverse-field quantum Ising model on a fractal structure, namely the Sierpinski carpet. When a magnetic field Δ is applied perpendicular to the Ising spin direction, quantum fluctuations affect the transition between the ferromagnetic and the paramagnetic phases. Employing the continuous-time quantum Monte Carlo simulation method and the finite-size scaling analysis, we investigate the interplay between the quantum fluctuations and the exotic dimensionality of the fractal structure and its effect on the critical behavior. As the transverse magnetic field increases, the critical temperature monotonically decreases until it apparently vanishes at a critical field Δ(c), beyond which the system becomes paramagnetic at all temperatures. However, the critical exponents are independent of Δ and remain the same as in the purely classical(Δ=0) case.
The dynamic phase transition (DPT) temperatures and the dynamic phase diagrams are obtained for spin-1/2 Ising system under the presence of a time-dependent oscillating external magnetic field using the method that was purposed by Meijer and Edwards [ANNALS of PHYSICS, 54, 240 (1969)] within the framework of the Bethe or constant coupling theory. The time variation of magnetization is investigated to find the phases in the system. Thermal behavior of the average magnetization is studied to characterize the nature (continuous and discontinuous) of transitions and to obtain the dynamic phase transition temperatures. The DPT temperatures are found always a second-order; hence there can be no tricritical point separating lines of first- and second-order dynamic phase transitions. This result agrees with the dynamic Monte-Carlo (MC) simulations of a two-dimensional kinetic spin-1/2 Ising model in an oscillating external magnetic field.
We investigate the crossover of critical behavior for the dynamic phase transition (DPT) in ferromagnetic thin films using Monte Carlo simulations of the kinetic Ising model, focusing on the scaling behavior of the dynamic order parameter under a time-dependent external magnetic field. Specifically, we study the transition of the critical behavior of such multilayer film systems from two-dimensional (2D) to three-dimensional (3D) as a function of the film thickness and the distance to the critical point, which enables dimensional crossover observations. Our results indicate that the effective critical exponents exhibit a clear transition in their scaling behavior, with thinner films showing 2D-like characteristics and thicker films displaying 3D-like behavior, for both the DPT and the thermodynamic phase transitions (TPT). Quantitatively, the crossover from 2D to 3D behavior occurs at larger film thicknesses for the DPT compared to the TPT, suggesting that DPT and TPT are governed by distinctly different length scales and underlying surface effects. These findings are in agreement with experimental observations in ultrathin Co films, where dynamic and thermodynamic critical exponents were found to differ. Therefore, our study provides an in-depth explanation for critical phenomena in thin-film ferromagnets driven by a time-dependent magnetic field. By comparing the dimensional crossover properties of both TPT and DPT, we present a comprehensive understanding of how thin-film geometry and surface effects influence the scaling laws and critical behavior in nonequilibrium systems.
Using Monte Carlo simulations based on the Metropolis algorithm, we investigate the dynamic phase transition properties of kinetic Ising model driven by a sinusoidally oscillating magnetic field in the presence of additive white noise. We calculate equilibrium and dynamic properties such as the temperature dependence of average magnetization and magnetic specific heat, as well as the period dependence of dynamic order parameter and scaled variance. After determining the critical period at which order-disorder transition takes place, we perform finite size scaling analysis to extract the exponent ratios, and discuss the variation of these properties in the presence of noisy magnetic field. As a general result, we show that for a noisy system, DPT does not fall into a universality class of the conventional dynamic (and also equilibrium) universality class of the Ising model.
We calculate the dynamic phase transition (DPT) temperatures and present the dynamic phase diagrams in the kinetic mixed spin-1/2 and spin-5/2 Ising model under the presence of a time-dependent oscillating external magnetic field. We employ the Glauber transition rates to construct the set of mean-field dynamic equations. The time variation of the average magnetizations and the thermal behavior of the dynamic magnetizations are investigated, extensively. The nature (continuous or discontinuous) of the transitions is characterized by studying the thermal behaviors of the dynamic magnetizations. The DPT points are obtained and the phase diagrams are presented in two different planes. Phase diagrams contain four fundamental phases and three coexistence or mixed phases, which strongly depend on interaction parameters. The phase diagrams are discussed and a comparison is made with the results of the other mixed spin Ising systems.