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arXiv:0806.1068 · Physical Review B

Noise spectra of stochastic pulse sequences: Application to large-scale magnetization flips in the finite size two-dimensional Ising model

Zhi Chen, Clare C. Yu

Published Apr 16, 2009• 4 citations
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Abstract

We provide a general scheme to predict and derive the contribution to the noise spectrum of a stochastic sequence of pulses from the distribution of pulse parameters. An example is the magnetization noise spectra of a two-dimensional Ising system near its phase transition. At $T\ensuremath{\le}{T}_{c}$, the low-frequency spectra is dominated by magnetization flips of nearly the entire system. We find that both the predicted and the analytically derived spectra fit those produced from simulations. Subtracting this contribution at ${T}_{c}$ leaves the high-frequency spectra which follow a power law determined by the critical exponents.

DOI
10.1103/physrevb.79.144420

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We provide a general scheme to predict and derive the contribution to the noise spectrum of a stochastic sequence of pulses from the distribution of pulse parameters. An example is the magnetization noise spectra of a two-dimensional Ising system near its phase transition. At T\ensuremath≤TcT\ensuremath{\le}{T}_{c}T\ensuremath≤Tc​, the low-frequency spectra is dominated by magnetization flips of nearly the entire system. We find that both the predicted and the analytically derived spectra fit those produced from simulations. Subtracting this contribution at Tc{T}_{c}Tc​ leaves the high-frequency spectra which follow a power law determined by the critical exponents.
We study the energy and magnetization noise spectra associated with first and second order phase transitions by using Monte Carlo simulations of the Ising model and 5-state Potts model in 2D. For a finite size system, the total noise power and the low frequency white noise S(f < fknee) increase as Tc is approached. In the thermodynamic limit S(f < fknee) diverges but fknee → 0 and the total noise power vanishes. f-1knee is approximately the equilibration time. At high frequencies S(f > fknee) ~ f-μ. For the Ising model, we relate μ to the critical exponents.
We consider power spectra for fluctuating quantities at a critical point, P( omega ) varies as omega - phi , where phi is determined by the dynamic exponent z and static exponents specific to the quantity considered. From the magnetization spectra obtained with simulations, z is found to be 2.13+or-0.03 for the two-dimensional Glauber kinetic Ising model, hence in accord with recent Monte Carlo relaxational studies, but at odds with those obtained by series expansion and damage spreading methods. Excellent dynamic finite-size scaling for system sizes L=10 to 128 supports our estimate.
We report the results of mean field and the Monte Carlo study of the dynamic magnetization-reversal transition in the Ising model, brought about by the application of an external field pulse applied in opposition to the existing order before the application of the pulse. The parameters controlling the transition are the strength hp{h}_{p}hp​ and the duration $\ensuremath{\Delta}t$ of the pulse. In the mean field case, an approximate analytical expression is obtained for the phase boundary which agrees well with that obtained numerically in the small $\ensuremath{\Delta}t$ and large TTT limit. The order parameter of the transition has been identified, and is observed to vary continuously near the transition. The order parameter exponent $\ensuremath{\beta}$ was estimated both for the mean field $(\ensuremath{\beta}=1)$ and the Monte Carlo $(\ensuremath{\beta}=0.90\ifmmode\pm\else\textpm\fi{}0.02$ in two dimensions) cases. The transition shows a ``critical slowing down'' type behavior near the phase boundary with diverging relaxation time. The divergence was found to be logarithmic in the mean field case and exponential in the Monte Carlo case. The finite size scaling technique was employed to estimate the correlation length exponent $\ensuremath{\nu}$ $(=1.5\ifmmode\pm\else\textpm\fi{}0.3$ in two dimensions) in the Monte Carlo case.
We propose that a maximum in measurement noise can be used as a signature of a phase transition. As an example, we study the energy and magnetization noise spectra associated with first- and second-order phase transitions by using Monte Carlo simulations of the Ising model and 5-state Potts model in two dimensions. For a finite size system, the total noise power and the low frequency white noise S(f<f(knee)) increase as T(c) is approached. In the thermodynamic limit, S(f<f(knee)) diverges but f(knee)-->0 and the total noise power vanishes. f(-1)(knee) is approximately the equilibration time.
We introduce a novel variance-reducing Monte Carlo algorithm for accurate determination of correlation times. We apply this method to two-dimensional Ising systems with sizes up to $15\ifmmode\times\else\texttimes\fi{}15$, using single-spin flip dynamics, random site selection, and transition probabilities according to the heat-bath method. From a finite-size scaling analysis of these correlation times, the dynamic critical exponent zzz is determined as z=2.1665(12)z\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2.1665(12)z=2.1665(12).
Abstract In undergraduate statistical mechanics courses the Ising model always plays an important role because it is the simplest non-trivial model used to describe magnetic systems. The one-dimensional model is easily solved analytically, while the two-dimensional one can be solved exactly by the Onsager solution. For this reason, numerical simulations are usually used to solve the two-dimensional model. Keeping in mind that the two-dimensional model is the platform for studying phase transitions, it is usually an exercise in computational undergraduate courses because its numerical solution is relatively simple to implement and its critical exponents are perfectly known. The purpose of this article is to present a detailed numerical study of the second-order phase transition in the two-dimensional Ising model at an undergraduate level, allowing readers not only to compare the mean-field solution, the exact solution and the numerical one through a complete study of the order parameter, the correlation function and finite-size scaling, but to present the techniques, along with hints and tips, for solving it themselves. We present the elementary theory of phase transitions and explain how to implement Markov chain Monte Carlo simulations and perform them for different lattice sizes with periodic boundary conditions. Energy, magnetization, specific heat, magnetic susceptibility and the correlation function are calculated and the critical exponents determined by finite-size scaling techniques. The importance of the correlation length as the relevant parameter in phase transitions is emphasized.