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Physical review. B, Condensed matter

Monte Carlo renormalization group study of the dynamic scaling of hysteresis in the two-dimensional Ising model

Fan Zhong

Published Aug 1, 2002• 36 citations
Publisher

Abstract

Extending the Monte Carlo renormalization group technique to the two-dimensional Ising model, we find from first principles that the dynamic scaling behavior of hysteresis originates from a rate exponent that characterizes the response of the system to the sweep rate of the field that leads to the hysteresis. The static hysteresis is determined and a scaling law attained. The effect of the temperature is discussed.

DOI
10.1103/physrevb.66.060401

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Journal of Physics Condensed Matter•Mar 2, 1998•16 citations
Extending the Monte Carlo renormalization group technique to the two-dimensional Ising model, we find from first principles that the dynamic scaling behavior of hysteresis originates from a rate exponent that characterizes the response of the system to the sweep rate of the field that leads to the hysteresis. The static hysteresis is determined and a scaling law attained. The effect of the temperature is discussed.
We extend the Monte Carlo renormalization-group technique to investigate the scaling behavior in the vicinity of the critical point of the two-dimensional Ising model in the presence of linearly changing temperatures, instead of some static temperatures. We find that it is possible in such a dynamically driven, nonequilibrium case to measure both the dynamic and static critical exponents. Two additional critical exponents associated with the temperature sweep rate are presented, and scaling laws relating them to other known exponents are obtained. Also a scaling form of the order parameter is extracted.
The critical dynamics of the two dimensional Kinetic Ising Model have been studied using Monte Carlo Real Space Renormalization Group techniques. Time dependent correlation functions were calculated for square lattices of 16 x 16 and 32 x 32 spins, using periodic boundary conditions wth 60,000 and 15,000 Monte Carlo steps per spin, respectively;The critical exponent z was estimated using a matching process for a variety of renormalization group operators. The estimates for z were surprisingly sensitive to small changes in the value of the renormalization group operator, with physically reasonable operators giving estimates for z less than the rigorous lower bound for z of z(,o) = 1.75. The estimates for z from a given renormalization group operator showed poor convergence properties. Attempts to estimate z from the scaling form of the correlation function were inconclusive, as were attempts to estimate the static exponent (eta);Possible explanations of these unexpected results are discussed. The most plausible source of the complex behavior seen in this work is the discrete nature of time in the Monte Carlo computer simulation, which generates phenomena not present in standard theoretical treatments of the Kinetic Ising Model.
We propose a simple method of Monte Carlo renormalization group. Thermodynamic quantities of the block-spin system are analyzed phenomenologically on the basis of scaling concept. The field exponent yH is obtained from the square of block-spin magnetization at the fixed point, while the temperature exponent yT is obtained as a consequence of nonlinearity of the scaling fields. In order to check the effectiveness of the method, we perform calculation on the three-dimensional Ising model. The critical temperature and the two critical exponents are estimated precisely. Finite-size effect is shown to be weak, whose origin is the regular part of the free energy density. Statistical errors of the ratios of the block-spin quantities are proved to be surprisingly small compared to those of the block-spin quantities themselves.
The critical relaxation of the magnetization in a two-dimensional Ising model with quenched random non-magnetic impurities has been studied by numerical simulation. A squared lattice of size 400 2 with spin concentrations p=1.0; 0.95, 0,9, 0.85, 0.8, 0.75 and 0.7 was considered. The dynamical critical exponent z was determined by the Monte Carlo method combined with the dynamical renormalization group method. The following values were received for z(p): z(1)=2.24+or-0.07, z(0.95)=2.24+or-0.06, z(0.9)=2.24+or-0.06, z(0.85)=2.38+or-0.05, z(0.8)=2.51+or-0.06, z(0.75)=2.66+or-0.07 and z(0.7)=2.88+or-0.06. A singular dynamic scaling behaviour with z=A' mod ln(p-p c ) mod +B' was found for systems with spin concentrations p<or=0.85 near the percolation threshold p c , where A'=0.56+or-0.07 and B'=1.62+or-0.07 are temperature-independent constants.
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.