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Journal of Physics Condensed Matter

Hysteresis scaling of the field-driven first-order phase transition in the Ising model

Guangping Zheng, J. X. Zhang

Published Mar 2, 1998• 16 citations
Publisher

Abstract

Dynamical phase transitions in the Ising model on hypercubic lattices are considered. Under a linearly swept magnetic field, the hysteresis loop that characterizes the field-driven first-order phase transition is studied carefully. Using the Glauber dynamics, we find that, in the mean-field approximation, the energy dissipation of this phase transition or the hysteresis loop area A of the M-H curve can be scaled with respect to the sweep rate h of magnetic field in the form with a = 2 and b = 2/3. However, b varies when fluctuations and spin correlations are taken into account. Monte Carlo simulation is used to obtain the scaling relation for A in two-, three- and four-dimensional Ising models and we obtain the exponents and respectively. These exponents are obviously different from those obtained by scaling A as for any temperatures in Ising models under a sinusoidal field. Finally we point out that, in the concept of universality, field-driven first-order phase transitions in the Ising model in different dimensions belong to different universal classes due to the spin fluctuation and correlation below the Curie temperature.

DOI
10.1088/0953-8984/10/8/018

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Dynamical phase transitions in the Ising model on hypercubic lattices are considered. Under a linearly swept magnetic field, the hysteresis loop that characterizes the field-driven first-order phase transition is studied carefully. Using the Glauber dynamics, we find that, in the mean-field approximation, the energy dissipation of this phase transition or the hysteresis loop area A of the M-H curve can be scaled with respect to the sweep rate h of magnetic field in the form with a = 2 and b = 2/3. However, b varies when fluctuations and spin correlations are taken into account. Monte Carlo simulation is used to obtain the scaling relation for A in two-, three- and four-dimensional Ising models and we obtain the exponents and respectively. These exponents are obviously different from those obtained by scaling A as for any temperatures in Ising models under a sinusoidal field. Finally we point out that, in the concept of universality, field-driven first-order phase transitions in the Ising model in different dimensions belong to different universal classes due to the spin fluctuation and correlation below the Curie temperature.
Abstract Phase transitions are divided into first-order and continuous ones in the current classification. While the latter shows striking phenomena of scaling and universality, the former is typically characterized by discontinuous jumps in extensive variables and pronounced hysteresis. Recent studies have demonstrated universal scaling behavior controlled by a cubic fixed point in first-order phase transitions. However, more recent investigations of hysteresis in a dynamic mean-field quartic model driven across its first-order phase transitions have revealed new scaling exponents for different driving rates. Here, we discover a new exponent for large driving rates arising surprisingly from critical phenomena and show that, depending on the magnitude of the driving rates and the absence or presence of noise, the same mean-field model remarkably exhibits several universality classes with definite universal scaling exponents governed by their corresponding fixed points, as revealed through a systematic scaling analysis based on renormalization group theory. The theories, the various crossovers between different fixed points, and the complete universal scaling of full curve collapse, are all verified by numerical results. This further confirms universal scaling in first-order phase transitions.
The universality class of the dynamic magnetization-reversal transition, induced by a competing field pulse, in an Ising model on a square lattice, below its static ordering temperature, is studied here using Monte Carlo simulations. Fourth-order cumulant of the order parameter distribution is studied for different system sizes around the phase boundary region. The crossing point of the cumulant (for different system sizes) gives the transition point and the value of the cumulant at the transition point indicates the universality class of the transition. The cumulant value at the crossing point for low temperature and pulse width range is observed to be significantly less than that for the static transition in the same two-dimensional Ising model. The finite-size scaling behavior in this range also indicates a higher correlation length exponent value. For higher temperature and pulse width range, the transition seems to fall in a mean-field-like universality class.
Hysteresis loops are often seen in experiments at first-order phase transformations, when the system goes out of equilibrium. They may have a macroscopic jump (roughly as in the supercooling of liquids) or they may be smoothly varying (as seen in most magnets). We have studied the nonequilibrium zero-temperature random-field Ising-model as a model for hysteretic behavior at first-order phase transformations. As disorder is added, one finds a transition where the jump in the magnetization (corresponding to an infinite avalanche) decreases to zero. At this transition we find a diverging length scale, power-law distributions of noise (avalanches), and universal behavior. We expand the critical exponents about mean-field theory in 6-\ensuremath{\epsilon} dimensions. Using a mapping to the pure Ising model, we Borel sum the 6-\ensuremath{\epsilon} expansion to O(${\mathrm{\ensuremath{\epsilon}}}^{5}$) for the correlation length exponent. We have developed a method for directly calculating avalanche distribution exponents, which we perform to O(\ensuremath{\epsilon}). Our analytical predictions agree with numerical exponents in two, three, four, and five dimensions [Perkovi\ifmmode \acute{c}\else \'{c}\fi{} et al., Phys. Rev. Lett. 75, 4528 (1995)]. \textcopyright{} 1996 The American Physical Society.
We study the scaling and universal behavior of temperature-driven first-order phase transitions in scalar models. These transitions are found to exhibit rich phenomena, though they are controlled by a single complex-conjugate pair of imaginary fixed points of ϕ 3 theory. Scaling theories and renormalization group theories are developed to account for the phenomena, and three universality classes with their own hysteresis exponents are found: a field-like thermal class, a partly thermal class, and a purely thermal class, designated, respectively, as Thermal Classes I, II, and III. The first two classes arise from the opposite limits of the scaling forms proposed and may cross over to each other depending on the temperature sweep rate. They are both described by a massless model and a purely massive model, both of which are equivalent and are derived from ϕ 3 theory via symmetry. Thermal Class III characterizes the cooling transitions in the absence of applied external fields and is described by purely thermal models, which include cases in which the order parameters possess different symmetries and thus exhibit different universality classes. For the purely thermal models whose free energies contain odd-symmetry terms, Thermal Class III emerges only at the mean-field level and is identical to Thermal Class II. Fluctuations change the model into the other two models. Using the extant three- and two-loop results for the static and dynamic exponents for the Yang–Lee edge singularity, respectively, which falls into the same universality class as ϕ 3 theory, we estimate the thermal hysteresis exponents of the various classes to the same precision. Comparisons with numerical results and experiments are briefly discussed.