Local Update Algorithms Research Articles

Parallel tempered metadynamics: Overcoming potential barriers without surfing or tunneling

At fine lattice spacings, Markov chain Monte Carlo simulations of QCD and other gauge theories with or without fermions are plagued by slow modes that give rise to large autocorrelation times. This can lead to simulation runs that are effectively stuck in one topological sector, a problem known as topological freezing. Here, we demonstrate that for a relevant set of parameters, metadynamics can be used to unfreeze four-dimensional SU(3) gauge theory. However, compared to local update algorithms and the Hybrid Monte Carlo algorithm, the computational overhead is significant in pure gauge theory, and the required reweighting procedure may considerably reduce the effective sample size. To deal with the latter problem, we propose modifications to the metadynamics bias potential and the combination of metadynamics with parallel tempering. We test the new algorithm in four-dimensional SU(3) gauge theory and find that it can achieve topological unfreezing without compromising the effective sample size, thereby reducing the autocorrelation times of topological observables by at least 2 orders of magnitude compared to conventional update algorithms. Additionally, we observe significantly improved scaling of autocorrelation times with the lattice spacing in two-dimensional U(1) gauge theory. Published by the American Physical Society 2024

General method to sample systems in the microcanonical ensemble using Monte Carlo simulations

Monte Carlo simulations have boosted the numerical study of several different physical systems and in particular, the canonical ensemble has been especially useful because of the existence of easy and efficient simulation algorithms. Nevertheless, nature does not know about statistical ensembles and therefore it is desirable and a theoretical challenge to show how to perform numerical simulations in the microcanonical ensemble without the use of unphysical degrees of freedom. In this article, we present a straightforward applicable method based on the concepts of a configurational temperature estimator (Rugh Phys Rev Lett 78:772, 1997; Gutierrez et al. J Phys A Math Theor 51:455003, 2018) and on stochastic dynamics, which is independent of the Monte Carlo update strategy, and can be implemented for both local update or cluster algorithms. We illustrate it by performing a numerical simulation of the two-dimensional XY-model, finding the equilibrium temperature of two spin subsystems initially at different temperatures when they are put into thermal contact.

Temperature scaling in nonequilibrium relaxation in three-dimensional Heisenberg model in the Swendsen-Wang and Metropolis algorithms.

Recently the present authors proposed the nonequilibrium-to-equilibrium scaling (NE-ES) scheme for the critical Monte Carlo relaxation process [Nonomura, J. Phys. Soc. Jpn. 83, 113001 (2014)10.7566/JPSJ.83.113001], which scales relaxation data in the whole simulation-time regions regardless of functional forms, namely, both for the stretched-exponential critical relaxation in cluster algorithms and for the power-law critical relaxation in local-update algorithms. In the present study, we generalize this scheme to off-critical relaxation process and scale relaxation data for various temperatures in the whole simulation-time regions. This proposal of the off-critical scaling in cluster algorithms cannot be described by the dynamical finite-size scaling theory based on the power-law critical relaxation. As an example, we investigate the three-dimensional classical Heisenberg model previously analyzed with the NE-ES [Nonomura and Tomita, Phys. Rev. E 93, 012101 (2016)10.1103/PhysRevE.93.012101] in the Swendsen-Wang and Metropolis algorithms.

Boosting Monte Carlo simulations of spin glasses using autoregressive neural networks.

The autoregressive neural networks are emerging as a powerful computational tool to solve relevant problems in classical and quantum mechanics. One of their appealing functionalities is that, after they have learned a probability distribution from a dataset, they allow exact and efficient sampling of typical system configurations. Here we employ a neural autoregressive distribution estimator (NADE) to boost Markov chain Monte Carlo (MCMC) simulations of a paradigmatic classical model of spin-glass theory, namely, the two-dimensional Edwards-Anderson Hamiltonian. We show that a NADE can be trained to accurately mimic the Boltzmann distribution using unsupervised learning from system configurations generated using standard MCMC algorithms. The trained NADE is then employed as smart proposal distribution for the Metropolis-Hastings algorithm. This allows us to perform efficient MCMC simulations, which provide unbiased results even if the expectation value corresponding to the probability distribution learned by the NADE is not exact. Notably, we implement a sequential tempering procedure, whereby a NADE trained at a higher temperature is iteratively employed as proposal distribution in a MCMC simulation run at a slightly lower temperature. This allows one to efficiently simulate the spin-glass model even in the low-temperature regime, avoiding the divergent correlation times that plague MCMC simulations driven by local-update algorithms. Furthermore, we show that the NADE-driven simulations quickly sample ground-state configurations, paving the way to their future utilization to tackle binary optimization problems.

Nonequilibrium-relaxation approach to quantum phase transitions: Nontrivial critical relaxation in cluster-update quantum Monte Carlo.

Although the nonequilibrium-relaxation (NER) method has been widely used in Monte Carlo studies on phase transitions in classical spin systems, such studies have been quite limited in quantum phase transitions. The reason is that the relaxation process based on cluster-update quantum Monte Carlo (QMC) algorithms, which are now standards in Monte Carlo studies on quantum systems, has been considered "too fast" for such analyses. Recently, the present authors revealed that the NER process in classical spin systems based on cluster-update algorithms is characterized by stretched-exponential critical relaxation, rather than conventional power-law relaxation in local-update algorithms. In the present article, we show that this is also the case in quantum phase transitions analyzed with the cluster-update QMC. As the simplest example of isotropic quantum spin models that exhibit quantum phase transitions, we investigate the Néel-dimer quantum phase transition in the two-dimensional S=1/2 columnar-dimerized antiferromagnetic Heisenberg model with the continuous-time loop algorithm, and we confirm stretched-exponential critical relaxation consistent with the three-dimensional classical Heisenberg model in the Swendsen-Wang algorithm.

Measuring the topological susceptibility in a fixed sector

For field theories with a topological charge Q, it is often of interest to measure the topological susceptibility chi_t = ( < Q^2 > - < Q >^2 ) / V. If we manage to perform a Monte Carlo simulation where Q changes frequently, chi_t can be evaluated directly. However, for local update algorithms and fine lattices, the auto-correlation time with respect to Q tends to be extremely long, which invalidates the direct approach. Nevertheless, the measurement of chi_t is still feasible, even when the entire Markov chain is topologically frozen. We test a method for this purpose, based on the correlation of the topological charge density, as suggested by Aoki, Fukaya, Hashimoto and Onogi. Our studies in non-linear sigma-models and in 2d Abelian gauge theory yield accurate results for chi_t, which confirm that the method is applicable. We also obtain promising results in 4d SU(2) Yang-Mills theory, which suggest the applicability of this method in QCD.

Topological susceptibility from slabs

In quantum field theories with topological sectors, a non-perturbative quantity of interest is the topological susceptibility chi_t. In principle it seems straightforward to measure chi_t by means of Monte Carlo simulations. However, for local update algorithms and fine lattice spacings, this tends to be difficult, since the Monte Carlo history rarely changes the topological sector. Here we test a method to measure chi_t even if data from only one sector are available. It is based on the topological charges in sub-volumes, which we denote as slabs. Assuming a Gaussian distribution of these charges, this method enables the evaluation of chi_t, as we demonstrate with numerical results for non-linear sigma-models.

First-order directional ordering transition in the three-dimensional compass model

We study the low-temperature properties of the classical three-dimensional compass or $t_{2g}$ orbital model on simple-cubic lattices by means of comprehensive large-scale Monte Carlo simulations. Our numerical results give evidence for a directionally ordered phase that is reached via a first-order transition at the temperature $T_0 = 0.098328(3) J / k_{\mathrm{B}}$. To obtain our results we employ local and cluster update algorithms, parallel tempering and multiple histogram reweighting as well as model-specific screw-periodic boundary conditions, which help counteract severe finite-size effects.

New Nonequilibrium-to-Equilibrium Dynamical Scaling and Stretched-Exponential Critical Relaxation in Cluster Algorithms

Nonequilibrium relaxation behaviors in the Ising model on a square lattice based on the Wolff algorithm are totally different from those based on local-update algorithms. In particular, the critical relaxation is described by the stretched-exponential decay. We propose a novel scaling procedure to connect nonequilibrium and equilibrium behaviors continuously, and find that the stretched-exponential scaling region in the Wolff algorithm is as wide as the power-law scaling region in local-update algorithms. We also find that relaxation to the spontaneous magnetization in the ordered phase is characterized by the exponential decay, not the stretched-exponential decay based on local-update algorithms.

SIMULATING SPIN MODELS ON GPU: A TOUR

The use of graphics processing units (GPUs) in scientific computing has gathered considerable momentum in the past five years. While GPUs in general promise high performance and excellent performance per Watt ratios, not every class of problems is equally well suitable for exploiting the massively parallel architecture they provide. Lattice spin models appear to be prototypic examples of problems suitable for this architecture, at least as long as local update algorithms are employed. In this review, I summarize our recent experience with the simulation of a wide range of spin models on GPU employing an equally wide range of update algorithms, ranging from Metropolis and heat bath updates, over cluster algorithms to generalized ensemble simulations.

Stripe-tetragonal phase transition in the two-dimensional Ising model with dipole interactions: Partition function zeros approach

We have performed multicanonical simulations to study the critical behavior of the two-dimensional Ising model with dipole interactions. This study concerns the thermodynamic phase transitions in the range of the interaction δ where the phase characterized by striped configurations of width h = 1 is observed. Controversial results obtained from local update algorithms have been reported for this region, including the claimed existence of a second-order phase transition line that becomes first order above a tricritical point located somewhere between δ = 0.85 and 1. Our analysis relies on the complex partition function zeros obtained with high statistics from multicanonical simulations. Finite size scaling relations for the leading partition function zeros yield critical exponents ν that are clearly consistent with a single second-order phase transition line, thus excluding such a tricritical point in that region of the phase diagram. This conclusion is further supported by analysis of the specific heat and susceptibility of the orientational order parameter.

PHASE TRANSITION IN THE ISING MODEL ON LOCAL-WORLD EVOLVING NETWORKS

We study the critical behavior of the Ising model on the local-world evolving network. Monte Carlo simulations with the standard Metropolis local update algorithms are performed extensively on the network with different parameters. Ising spins put onto network vertices exhibit an effective phase transition from ferromagnetism to paramagnetism upon heating. The critical temperature has been demonstrated to increase linearly with the average degree of the network as TC ~ 〈k〉. Simulation results on local-world evolving networks with various parameters show logarithmical relationships of the critical temperature with the size of the local world as TC ~ ln (ml), and with the size of the network as TC ~ ln (N), respectively. The latter is the generalization of the conclusion for the Ising model on the Barabási–Albert scale-free network, a limiting case of the local-world evolving network.

ModifiedSO(3)lattice gauge theory atT≠0with parallel tempering: Monopole and vortex condensation

The deconfinement transition is studied close to the continuum limit of $SO(3)$ lattice gauge theory. High barriers for tunneling among different twist sectors causing loss of ergodicity for local update algorithms are circumvented by means of parallel tempering. We compute monopole and center vortex free energies both within the confining phase and through the deconfinement transition. We discuss in detail the general problem of defining order parameters for adjoint actions.

Vortex free energy and deconfinement in center-blind discretizations of Yang-Mills theories

Maximal 't Hooft loops are studied in SO(3) lattice gauge theory at finite temperature T. Tunneling barriers among twist sectors causing loss of ergodicity for local update algorithms are overcome through parallel tempering, enabling us to measure the vortex free energy F and to identify a deconfinement transition at some $\beta_A^{crit}$. The behavior of F below $\beta_A^{crit}$ shows however striking differences with what is expected from discretizations in the fundamental representation.

Fast flat-histogram method for generalized spin models

We present a Monte Carlo method that efficiently computes the density of states for spin models having any number of interaction per spin. By combining a random walk in the energy space with collective updates controlled by the microcanonical temperature, our method yields dynamic exponents close to their ideal random-walk values, reduced equilibrium times, and very low statistical error in the density of states. The method can host any density of states estimation scheme, including the Wang-Landau algorithm and the transition matrix method. Our approach proves remarkably powerful in the numerical study of models governed by long-range interactions, where it is shown to reduce the algorithm complexity to that of a short-range model with the same number of spins. We apply the method to the -state Potts chains with power-law decaying interactions in their first-order regime; we find that conventional local-update algorithms are outperformed already for sizes above a few hundred spins. By considering chains containing up to spins, which we simulated in fairly reasonable time, we obtain estimates of transition temperatures correct to five-figure accuracy. Finally, we propose several efficient schemes aimed at estimating the microcanonical temperature.

A Fast and Robust Cluster Update Algorithm for Image Segmentation in Spin-Lattice Models Without Annealing. Visual Latencies Revisited.

Image segmentation in spin-lattice models relies on the fast and reliable assignment of correct labels to those groups of spins that represent the same object. Commonly used local spin-update algorithms are slow because in each iteration only a single spin is flipped and a careful annealing schedule has to be designed in order to avoid local minima and correctly label larger areas. Updating of complete spin clusters is more efficient, but often clusters that should represent different objects will be conjoined. In this study, we propose a cluster update algorithm that, similar to most local update algorithms, calculates an energy function and determines the probability for flipping a whole cluster of spins by the energy gain calculated for a neighborhood of the regarded cluster. The novel algorithm, called energy-based cluster update (ECU algorithm), is compared to its predecessors. A convergence proof is derived, and it is shown that the algorithm outperforms local update algorithms by far in speed and reliability. At the same time it is more robust and noise tolerant than other versions of cluster update algorithms, making annealing completely unnecessary. The reduction in computational effort achieved this way allows us to segment real images in about 1-5 sec on a regular workstation. The ECU-algorithm can recover fine details of the images, and it is to a large degree robust with respect to luminance-gradients across objects. In a final step, we introduce luminance dependent visual latencies (Opara and Worgotter, 1996; Worgotter, Opara, Funke, and Eysel, 1996) into the spin-lattice model. This step guarantees that only spins representing pixels with similar luminance become activated at the same time. The energy function is then computed only for the interaction of the regarded cluster with the currently active spins. This latency mechanism improves the quality of the image segmentation by another 40%. The results shown are based on the evaluation of gray-level differences. It is important to realize that all algorithmic components can be transferred easily to arbitrary image features, like disparity, texture, and motion.

Nonlocal Monte Carlo algorithms for statistical physics applications

After a brief general overview of Monte Carlo computer simulations in statistical physics, special emphasis is placed on applications to phase transitions and critical phenomena. Here, standard simulations employing local update algorithms are severely hampered by the problem of critical slowing down, that is by strong correlations between successively generated data. It is shown that this problem can be greatly reduced by using nonlocal update techniques such as cluster and multigrid algorithms. The general ideas are illustrated for simple lattice spin models and Euclidean path integrals.

Monte Carlo Algorithms for the Fully Frustrated XY Model

We investigate local update algorithms for the fully frustrated XY model on a square lattice. In addition to the standard updating procedures like the Metropolis or heat bath algorithm we include overrelaxation sweeps, implemented through single spin updates that preserve the energy of the configuration. The dynamical critical exponent (of order two) stays more or less unchanged. However, the integrated autocorrelation times of the algorithm can be significantly reduced.

Multicanonical Monte Carlo simulations

Canonical Monte Carlo simulations of disordered systems like spin glasses and systems undergoing first-order phase transitions are severely hampered by rare event states which lead to exponentially diverging autocorrelation times with increasing system size and hence to exponentially large statistical errors. One possibility to overcome this problem is the multicanonical reweighting method. Using standard local update algorithms it could be demonstrated that the dependence of autocorrelation times on the system size V is well described by a less divergent power law, τ ∝ V α , with 1< α<3, depending on the system. After a brief review of the basic ideas, combinations of multicanonical reweighting with non-local update algorithms will be discussed. With the multibondic algorithm, which combines multicanonical reweighting with cluster updates, the dynamical exponent α can be reduced to unity, the optimal value one would expect from a random walk argument. Asymptotically for large system sizes the multibondic algorithm therefore always performs better than the standard multicanonical method. Finally it is shown that a combination with multigrid update techniques improves the performance of multicanonical simulations by roughly one order of magnitude, uniformly for all system sizes.

Simulating very large Ising systems for short timescales

We present a new approach for investigating very large lattices of the Ising type with local update algorithms for short time scales. Instead of the conventional storing of configurations and iterating through time, we calculate spins for all time steps simultaneously and propagate through the lattice. We apply the algorithm to the two-dimensional Ising model with Metropolis dynamics and simulate systems of up to (4 × 10 5) 2 lattice sites for 10 MC steps.