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arXiv:2104.13264

Collective Monte Carlo updates through tensor network renormalization

Miguel Frías-Pérez, Michael Mariën, David Pérez García, Mari Carmen Bañuls, S. Iblisdir

Published Apr 27, 2021• 1 citation
arXivHTMLPDF

Abstract

We introduce a Metropolis-Hastings Markov chain for Boltzmann distributions of classical spin systems. It relies on approximate tensor network contractions to propose correlated collective updates at each step of the evolution. We present benchmarks for a wide variety of instances of the two-dimensional Ising model, including ferromagnetic, antiferromagnetic, (fully) frustrated and Edwards-Anderson spin glass cases, and we show that, with modest computational effort, our Markov chain achieves sizeable acceptance rates, even in the vicinity of critical points. In each of the situations we have considered, the Markov chain compares well with other Monte Carlo schemes such as the Metropolis or Wolff algorithm: equilibration times appear to be reduced by a factor that varies between 40 and 2000, depending on the model and the observable being monitored. We also present an extension to three spatial dimensions, and demonstrate that it exhibits fast equilibration for finite ferro and antiferromagnetic instances. Additionally, and although it is originally designed for a square lattice of finite degrees of freedom with open boundary conditions, the proposed scheme can be used as such, or with slight modifications, to study triangular lattices, systems with continuous degrees of freedom, matrix models, a confined gas of hard spheres, or to deal with arbitrary boundary conditions.

DOI
10.48550/arxiv.2104.13264

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We introduce a Metropolis-Hastings Markov chain for Boltzmann distributions of classical spin systems. It relies on approximate tensor network contractions to propose correlated collective updates at each step of the evolution. We present benchmarks for a wide variety of instances of the two-dimensional Ising model, including ferromagnetic, antiferromagnetic, (fully) frustrated and Edwards-Anderson spin glass cases, and we show that, with modest computational effort, our Markov chain achieves sizeable acceptance rates, even in the vicinity of critical points. In each of the situations we have considered, the Markov chain compares well with other Monte Carlo schemes such as the Metropolis or Wolff algorithm: equilibration times appear to be reduced by a factor that varies between 40 and 2000, depending on the model and the observable being monitored. We also present an extension to three spatial dimensions, and demonstrate that it exhibits fast equilibration for finite ferro and antiferromagnetic instances. Additionally, and although it is originally designed for a square lattice of finite degrees of freedom with open boundary conditions, the proposed scheme can be used as such, or with slight modifications, to study triangular lattices, systems with continuous degrees of freedom, matrix models, a confined gas of hard spheres, or to deal with arbitrary boundary conditions.
We introduce a Metropolis-Hastings Markov chain for Boltzmann distributions of classical spin systems. It relies on approximate tensor network contractions to propose correlated collective updates at each step of the evolution. We present benchmark computations for a wide variety of instances of the two-dimensional Ising model, including ferromagnetic, antiferromagnetic, (fully) frustrated and Edwards-Anderson spin glass instances, and we show that, with modest computational effort, our Markov chain achieves sizeable acceptance rates, even in the vicinity of critical points. In each of the situations we have considered, the Markov chain compares well with other Monte Carlo schemes such as the Metropolis or Wolff’s algorithm: equilibration times appear to be reduced by a factor that varies between 40 40 and 2000 2000 , depending on the model and the observable being monitored. We also present an extension to three spatial dimensions, and demonstrate that it exhibits fast equilibration for finite ferro- and antiferromagnetic instances. Additionally, and although it is originally designed for a square lattice of finite degrees of freedom with open boundary conditions, the proposed scheme can be used as such, or with slight modifications, to study triangular lattices, systems with continuous degrees of freedom, matrix models, a confined gas of hard spheres, or to deal with arbitrary boundary conditions.
Sampling equilibrium configurations of three-dimensional (3D) spin glasses with quenched disorder remains a fundamental challenge in statistical physics. The rugged energy landscape, pronounced critical slowing down, and intrinsic ergodicity breaking render standard Monte Carlo methods severely inefficient, particularly for large systems at low temperatures. In this Letter, we introduce the tensor-network Markov-chain Monte Carlo (TNMCMC) approach to address the issue. By formulating large-scale collective updates via tensor-network contractions on 2D lattice slices, TNMCMC yields unbiased samples of the Boltzmann distribution while substantially reducing autocorrelation times relative to conventional MCMC approaches. Numerical benchmarks on 3D spin glasses demonstrate that the method systematically mitigates critical slowing down. Furthermore, application to three- and four-state 3D Potts models shows that TNMCMC can traverse the exponential free-energy barriers of first-order phase transitions that typically hinder local MC updates. These results establish TNMCMC as a promising numerical framework capable of addressing important 3D problems in statistical physics and beyond, such as the precise computation of low-temperature thermodynamic properties of three-dimensional spin glasses and the decoding of quantum error-correcting codes with a three-dimensional topology.
Disordered lattice spin systems are crucial in both theoretical and applied physics. However, understanding their properties poses significant challenges to Monte Carlo simulations. In this work, we investigate the two-dimensional random-bond Ising model using the recently proposed tensor network Monte Carlo (TNMC) method. This method generates biased samples from conditional probabilities computed via tensor network contractions and corrects the bias using the Metropolis scheme. Consequently, the proposals provided by tensor networks function as block updates for Monte Carlo simulations. Through extensive numerical experiments, we demonstrate that TNMC simulations can be performed on lattices as large as $1024\ifmmode\times\else\texttimes\fi{}1024$ spins with moderate computational resources, a substantial increase from the previous maximum size of $64\ifmmode\times\else\texttimes\fi{}64$ in Markov-chain Monte Carlo. Notably, we observe an almost complete absence of critical slowing down, enabling the efficient collection of unbiased samples and averaging over a large number of random realizations of bond disorders. We successfully pinpoint the multicritical point along the Nishimori line with significant precision and accurately determined the bulk and surface critical exponents. Our findings suggest that TNMC is a highly efficient algorithm for exploring disordered and frustrated systems in two dimensions.
We present a method for Monte Carlo sampling on Ising spin systems, introducing a prior on the candidate moves in a Metropolis-Hastings scheme which can significantly reduce the rejection rate. The method can be regarded as an modification of the rejection-free method of Bortz, Kalos and Lebowitz (BKL) [Bortz et al. 1975], which was developed under two assumptions: i) low connectivity, and ii) that the energy shift induced by flipping one spin can only assume a small number of values. Our method employs the same probability distribution for the choice of the moves as BKL, but uses it as a prior in an otherwise standard Metropolis scheme: it is thus not fully rejection-free, but in a wide range of scenarios it is nearly so; moreover, it is easily extensible to more general situations which cannot be treated with the BKL method, in particular when the energy of the system can be written as the sum of two components, only one of which satisfies the assumptions at the basis of the BKL method. We provide two families of examples of such situations: graphs with heterogeneous continuous-valued couplings, and Monte Carlo simulations of quantum spin models in presence of a transverse field in the Suzuki-Trotter formalism. Our code is publicly available [this https URL], and extensible to user-defined models.
Efficient sampling of two-dimensional statistical physics systems remains a central challenge in computational statistical physics. Traditional Markov chain Monte Carlo (MCMC) methods, including cluster algorithms, provide only partial solutions, as their efficiency collapses for large systems in the presence of frustration and quenched disorder. The recently proposed Tensor Network Monte Carlo (TNMC) method offers a promising alternative, yet its original implementation suffers from inefficiencies due to the lack of scalable parallel sampling. In this work, we introduce BatchTNMC, a GPU-optimized and parallelized implementation of TNMC tailored for large-scale simulations of two-dimensional spin glasses. By leveraging batch processing and parallel sampling across multiple disorder realizations, our implementation achieves speedups of up to five orders of magnitude compared with the original serial scheme. Benchmarking on two-dimensional spin glasses demonstrates dramatic gains in efficiency: for instance, on a single GPU, BatchTNMC concurrently produces 1000 uncorrelated and unbiased samples across 1000 disorder realizations on 1024×10241024\times 10241024×1024 lattices in just 3.3 hours, with an acceptance probability of 37%. These results establish BatchTNMC as a scalable and powerful computational framework for the study of two-dimensional disordered spin glass systems.
Markov Chain Monte Carlo (MCMC), and Tensor Networks (TN) are two powerful frameworks for numerically investigating many-body systems, each offering distinct advantages. MCMC, with its flexibility and theoretical consistency, is well-suited for simulating arbitrary systems by sampling. TN, on the other hand, provides a powerful tensor-based language for capturing the entanglement properties intrinsic to many-body systems, offering a universal representation of these systems. In this work, we leverage the computational strengths of TN to design a versatile cluster MCMC sampler. Specifically, we propose a general framework for constructing tensor-based cluster MCMC methods, enabling arbitrary cluster updates by utilizing TNs to compute the distributions required in the MCMC sampler. Our framework unifies several existing cluster algorithms as special cases and allows for natural extensions. We demonstrate our method by applying it to the simulation of the two-dimensional Edwards-Anderson Model and the three-dimensional Ising Model. This work is dedicated to the memory of Prof. David Draper.