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arXiv:2007.14056 · Journal of Physics A Mathematical and Theoretical

Two-size probability-changing cluster algorithm

Tasrief Surungan, Yutaka Okabe

Published Nov 3, 2020• 0 citations
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Abstract

Abstract We propose a self-adapted Monte Carlo approach to automatically determine the critical temperature by simulating two systems with different sizes at the same temperature. The temperature is increased or decreased by checking the short-time average of the correlation ratios of the two system sizes. The critical temperature is achieved using the negative feedback mechanism, which can be regarded as an Ehrenfest model for diffusion with a central force. Moreover, the thermal average near the critical temperature can be calculated precisely. The proposed approach is a general method to treat second-order phase transition, first-order phase transition, and Berezinskii–Kosterlitz–Thouless transition on the equal footing.

DOI
10.1088/1751-8121/abc72c

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Abstract We propose a self-adapted Monte Carlo approach to automatically determine the critical temperature by simulating two systems with different sizes at the same temperature. The temperature is increased or decreased by checking the short-time average of the correlation ratios of the two system sizes. The critical temperature is achieved using the negative feedback mechanism, which can be regarded as an Ehrenfest model for diffusion with a central force. Moreover, the thermal average near the critical temperature can be calculated precisely. The proposed approach is a general method to treat second-order phase transition, first-order phase transition, and Berezinskii–Kosterlitz–Thouless transition on the equal footing.
We study the finite-size scaling (FSS) property of the correlation ratio, the ratio of the correlation functions with different distances. It is shown that the correlation ratio is a good estimator to determine the critical point of the second-order transition using the FSS analysis. The correlation ratio is especially useful for the analysis of the Kosterlitz-Thouless (KT) transition. We also present a generalized scheme of the probability-changing cluster algorithm, which has been recently developed by the present authors, based on the FSS property of the correlation ratio. We investigate the two-dimensional spin-12\frac{1}{2}21​ quantum XY\mathrm{XY}XY model of with this generalized scheme, obtaining the precise estimate of the KT transition temperature with less numerical effort.
A generalization to the quantum case of a recently introduced algorithm [Tomita and Okabe, Phys. Rev. Lett. 86, 572 (2001)] for the determination of the critical temperature of classical spin models is proposed. We describe a simple method to automatically locate critical points in (Quantum) Monte Carlo simulations. The algorithm assumes the existence of a finite correlation length in at least one of the two phases surrounding the quantum critical point. We illustrate these ideas on the example of the critical interchain coupling for which coupled antiferromagnetic S=1S=1S=1 spin chains order at T=0.T=0.T=0. Finite-size scaling relations are used to determine the exponents, $\ensuremath{\nu}=0.72(2)$ and $\ensuremath{\eta}=0.038(3)$ in agreement with previous estimates.
We test an improved finite-size scaling method for reliably extracting the critical temperature T BKT of a Berezinskii–Kosterlitz–Thouless (BKT) transition. Using known single-parameter logarithmic corrections to the spin stiffness ρ s at T BKT in combination with the Kosterlitz–Nelson relation between the transition temperature and the stiffness, ρ s ( T BKT ) = 2 T BKT /π, we define a size-dependent transition temperature T BKT ( L 1 , L 2 ) based on a pair of system sizes L 1 , L 2 , e.g., L 2 = 2 L 1 . We use Monte Carlo data for the standard two-dimensional classical XY model to demonstrate that this quantity is well behaved and can be reliably extrapolated to the thermodynamic limit using the next expected logarithmic correction beyond the ones included in defining T BKT ( L 1 , L 2 ). For the Monte Carlo calculations we use GPU (graphical processing unit) computing to obtain high-precision data for L up to 512. We find that the sub-leading logarithmic corrections have significant effects on the extrapolation. Our result T BKT = 0.8935(1) is several error bars above the previously best estimates of the transition temperature, T BKT ≈ 0.8929. If only the leading log-correction is used, the result is, however, consistent with the lower value, suggesting that previous works have underestimated T BKT because of the neglect of sub-leading logarithms. Our method is easy to implement in practice and should be applicable to generic BKT transitions.
Monte Carlo simulation with Metropolis algorithm has been performed on a cubic Ising lattice with four-spin couplings. Temperature dependence of the spin average and average energy has been obtained. We found that the phase transition is of first order when the strength of four-spin coupling is stronger a critical value, and this critical value is of lattice size dependence. Binder cumulants still can be used to determine the second order phase transition temperature. For first order phase transition, the spin average disappears abruptly at the temperature of the average energy being a third of the ground state energy.
Phase transitions and critical phenomena in several spin models are studied. These models include a two and three dimensional XY model and Ising models in structured and random field. The methods of study are cluster Monte Carlo simulations and network flow optimization algorithms. The XY model and a related soft spin model belonging to the O(2) universality class are studied using a modified version of the Invaded Cluster algorithm. The static critical properties of the models and the dynamic characteristics of the algorithm are studied. The critical exponent η for the O(2) universality class in two and three dimensions is measured. The Invaded Cluster algorithm for the XY model does not show any critical slowing for global variables. Benzene adsorption in a nanoporous solid is modeled with an Ising model with both internal and external magnetic fields. The first order phase transition and the critical properties of the system are studied. The system is simulated using the two replica cluster algorithm. The critical temperature of the system is found to decrease with increasing difference of the values of the internal fields. It is found that the system undergoes a phase transition for a relatively wide, experimentally obtainable, region of values of its parameters. The random field Ising model in three dimensions is studied at zero temperature. The nature of randomness induced phase transition is studied using a numerical network flow optimization method. It is found that the latent heat at the phase transition vanishes in the limit of infinite system size which can be taken as a proof of a second order phase transition.
Based on the short-time dynamic scaling form, a dynamic approach is proposed to numerically tackle the Kosterlitz-Thouless phase transition. Taking the two-dimensional XY\mathrm{XY}XY model as an example, the exponential divergence of the spatial correlation length, the transition temperature TKT,{T}_{\mathrm{KT}},TKT​, and all critical exponents are computed. Compared with Monte Carlo simulations in equilibrium, we obtain data at temperatures closer to TKT.{T}_{\mathrm{KT}}.TKT​.