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Register a new userParallel-tempering cluster algorithm for computer simulations of critical phenomena
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In finite-size scaling analyses of Monte Carlo simulations of second-order phase transitions one often needs an extended temperature range around the critical point. By combining the parallel tempering algorithm with cluster updates and an adaptive routine to find the temperature window of interest, we introduce a flexible and powerful method for systematic investigations of critical phenomena. As a result, we gain one to two orders of magnitude in the performance for 2D and 3D Ising models in comparison with the recently proposed Wang-Landau recursion for cluster algorithms based on the multibondic algorithm, which is already a great improvement over the standard multicanonical variant.
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We introduce a new update scheme to systematically improve the efficiency of parallel tempering simulations. We show that by adapting the number of sweeps between replica exchanges to the canonical autocorrelation time, the average round-trip time of a replica in temperature space can be significantly decreased. The temperatures are not dynamically adjusted as in previous attempts but chosen to yield a 50% exchange rate of adjacent replicas. We illustrate the new algorithm with results for the Ising model in two and the Edwards-Anderson Ising spin glass in three dimensions
We review several parallel tempering schemes and examine their main ingredients for accuracy and efficiency. The present study covers two selection methods of temperatures and several choices for the exchange of replicas, including a recent novel all-pair exchange method. We compare the resulting schemes and measure specific heat errors and efficiency using the two-dimensional (2D) Ising model. Our tests suggest that, an earlier proposal for using numbers of local moves related to the canonical correlation times is one of the key ingredients for increasing efficiency, and protocols using cluster algorithms are found to be very effective. Some of the protocols are also tested for efficiency and ground state production in 3D spin glass models where we find that, a simple nearest-neighbor approach using a local n-fold way algorithm is the most effective. Finally, we present evidence that the asymptotic limits of the ground state energy for the isotropic case and that of an anisotropic case of the 3D spin-glass model are very close and may even coincide.
We propose a new method for the determination of the weight factor for the simulated tempering method. In this method a short replica-exchange simulation is performed and the simulated tempering weight factor is obtained by the multiple-histogram reweighting techniques. The new algorithm is particularly useful for studying frustrated systems with rough energy landscape where the determination of the simulated tempering weight factor by the usual iterative process becomes very difficult. The effectiveness of the method is illustrated by taking an example for protein folding.
Recently, a hybrid percolation transitions (HPT) that exhibits both a discontinuous transition and critical behavior at the same transition point has been observed in diverse complex systems. In spite of considerable effort to develop the theory of HPT, it is still incomplete, particularly when the transition is induced by cluster merging dynamics. Here, we aim to develop a theoretical framework of the HPT induced by such dynamics. We find that two correlation-length exponents are necessary for characterizing the giant cluster and finite clusters, respectively. Finite-size scaling method for the HPT is also introduced. The conventional formula of the fractal dimension in terms of the critical exponents is not valid. Neither the giant nor finite clusters are fractals but they have fractal boundaries.
We applied the simulated tempering and magnetizing (STM) method to the two-dimensional three-state Potts model in an external magnetic field in order to perform further investigations of the STMs applicability. The temperature as well as the external field are treated as dynamical variables updated during the STM simulations. After we obtained adequate information for several lattice sizes $L$ (up to $160times 160$), we also performed a number of conventional canonical simulations of large lattices, especially in order to illustrate the crossover behavior of the Potts model in external field with increasing $L$. The temperature and external field for larger lattice size simulations were chosen by extrapolation of the detail information obtained by STM. We carefully analyzed the crossover scaling at the phase transitions with respect to the lattice size as well as the temperature and external field. The crossover behavior is clearly observed in the simulations in agreement with theoretical predictions.
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