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arXiv:1804.02709 · Physical review. E

Machine learning of phase transitions in the percolation and XY models

Wanzhou Zhang, Jiayu Liu, Tzu-Chieh Wei

Published Mar 29, 2019• 114 citations
arXivHTMLPDF

Abstract

In this paper, we apply machine learning methods to study phase transitions in certain statistical mechanical models on the two-dimensional lattices, whose transitions involve nonlocal or topological properties, including site and bond percolations, the XY model, and the generalized XY model. We find that using just one hidden layer in a fully connected neural network, the percolation transition can be learned and the data collapse by using the average output layer gives correct estimate of the critical exponent ν. We also study the Berezinskii-Kosterlitz-Thouless transition, which involves binding and unbinding of topological defects, vortices and antivortices, in the classical XY model. The generalized XY model contains richer phases, such as the nematic phase, the paramagnetic and the quasi-long-range ferromagnetic phases, and we also apply machine learning method to it. We obtain a consistent phase diagram from the network trained with only data along the temperature axis at two particular parameter Δ values, where Δ is the relative weight of pure XY coupling. Aside from using the spin configurations (either angles or spin components) as the input information in a convolutional neural network, we devise a feature engineering approach using the histograms of the spin orientations in order to train the network to learn the three phases in the generalized XY model and demonstrate that it indeed works. The trained network by using system size L×L can be used to the phase diagram for other sizes (L^{'}×L^{'}, where L^{'}≠L) without any further training.

DOI
10.1103/physreve.99.032142

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In this paper, we apply machine learning methods to study phase transitions in certain statistical mechanical models on the two-dimensional lattices, whose transitions involve nonlocal or topological properties, including site and bond percolations, the XY model, and the generalized XY model. We find that using just one hidden layer in a fully connected neural network, the percolation transition can be learned and the data collapse by using the average output layer gives correct estimate of the critical exponent ν. We also study the Berezinskii-Kosterlitz-Thouless transition, which involves binding and unbinding of topological defects, vortices and antivortices, in the classical XY model. The generalized XY model contains richer phases, such as the nematic phase, the paramagnetic and the quasi-long-range ferromagnetic phases, and we also apply machine learning method to it. We obtain a consistent phase diagram from the network trained with only data along the temperature axis at two particular parameter Δ values, where Δ is the relative weight of pure XY coupling. Aside from using the spin configurations (either angles or spin components) as the input information in a convolutional neural network, we devise a feature engineering approach using the histograms of the spin orientations in order to train the network to learn the three phases in the generalized XY model and demonstrate that it indeed works. The trained network by using system size L×L can be used to the phase diagram for other sizes (L^{'}×L^{'}, where L^{'}≠L) without any further training.
In recent years Machine Learning has proved to be successful in many technological applications and scientific tasks, such as image and speech recognition, natural language understanding and more. In this work we apply both supervised and unsupervised Machine Learning to three key problems in statistical physics: design of Hamiltonian from data, phase recognition and study of critical properties of a system undergoing phase transition. As an example of model design, we use a dataset with spin configurations and corresponding energies randomly sampled from a one dimensional Ising model. We then make two guesses for the correct Hamiltonian: the former involving interactions of spins with an external local field, the latter involving two body interactions among spins. The coupling constants are determined with a linear penalized regression, comparing the effects of L1 and L2 penalization terms of the cost function. We pay specific attention to the problem of overfitting and to the validation process, which is critical for accepting or rejecting the proposed model. The phase recognition problem is faced with the two dimensional Ising and XY models as examples. After showing the limits of a simple softmax regression for this task, we build suitable neural networks to overcome these limits. In particular, a feed forward network is built and the learning process is investigated for the Ising model; while a more sophisticated convolutional network is proposed for the XY model in order to detect local topological structures. The last part of the work is dedicated to the unsupervised study of phase transitions, and the determination of critical properties (order parameter, critical temperature, critical exponents). The discussed techniques are Principal Component Analysis (PCA) for dimensional reduction and K-means clustering for organizing data into subsets with specific properties. PCA is applied to the two dimensional Ising, Potts and XY models, and it is used to find suitable order parameters. The study of the proposed order parameter as a function of temperature provides evidence of phase transition, and a finite size scaling allows to extrapolate both the critical temperature in the thermodynamic limit, and the ν\nuν critical exponent for the correlation length. K-means clustering is applied to equilibrium configurations of the two dimensional Ising model, before and after the dimensional reduction, and the critical temperature is estimated. Moreover, a clustering of relaxation curves of magnetization in a Monte Carlo dynamics is used to build a phase diagram on the parameter space of temperature and magnetic field. A synergy of C++, Python and Wolfram Mathematica 12.0 is used throughout this work in order to sample input datasets and to build and control customized neural networks and learning tools. The most relevant codes are provided in the appendix.
We propose two machine-learning methods based on neural networks, which we respectively call the phase-classification method and the temperature-identification method, for detecting different types of phase transitions in the XXZ models without prior knowledge of their critical temperatures. The XXZ models have exchange couplings which are anisotropic in the spin space where the strength is represented by a parameter $\mathrm{\ensuremath{\Delta}}(>0)$. The models exhibit second-order phase transition when $\mathrm{\ensuremath{\Delta}}>1$, but the Berezinskii-Kosterlitz-Thouless (BKT) phase transition when $\mathrm{\ensuremath{\Delta}}<1$. In the phase-classification method, the neural network is trained using spin or vortex configurations of well-known classical spin models other than the XXZ models, e.g., the Ising models and the XY models, to classify those of the XXZ models to corresponding phases. We demonstrate that the trained neural network successfully detects the phase transitions for both $\mathrm{\ensuremath{\Delta}}>1$ and $\mathrm{\ensuremath{\Delta}}<1$, and the evaluated critical temperatures coincide well with those evaluated by conventional numerical calculations. In the temperature-identification method, on the other hand, the neural network is trained so as to identify temperatures at which the input spin or vortex configurations are generated by the Monte Carlo thermalization. The critical temperatures are evaluated by analyzing the optimized weight matrix, which coincides with the result of numerical calculation for the second-order phase transition in the Ising-like XXZ model with $\mathrm{\ensuremath{\Delta}}=1.05$ but cannot be determined uniquely for the BKT transition in the XY-like XXZ model with $\mathrm{\ensuremath{\Delta}}=0.95$.
We present an analysis of neural network-based machine learning schemes for phases and phase transitions in theoretical condensed matter research, focusing on neural networks with a single hidden layer. Such shallow neural networks were previously found to be efficient in classifying phases and locating phase transitions of various basic model systems. In order to rationalize the emergence of the classification process and for identifying any underlying physical quantities, it is feasible to examine the weight matrices and the convolutional filter kernels that result from the learning process of such shallow networks. Furthermore, we demonstrate how the learning-by-confusing scheme can be used, in combination with a simple threshold-value classification method, to diagnose the learning parameters of neural networks. In particular, we study the classification process of both fully-connected and convolutional neural networks for the two-dimensional Ising model with extended domain wall configurations included in the low-temperature regime. Moreover, we consider the two-dimensional XY model and contrast the performance of the learning-by-confusing scheme and convolutional neural networks trained on bare spin configurations to the case of preprocessed samples with respect to vortex configurations. We discuss these findings in relation to similar recent investigations and possible further applications.
Using a supervised neural network (NN) trained once on a one-dimensional lattice of 200 sites, we calculate the Berezinskii–Kosterlitz–Thouless phase transitions of the two-dimensional (2D) classical XY and the 2D generalized classical XY models. In particular, both the bulk quantities Binder ratios and the spin states of the studied systems are employed to construct the needed configurations for the NN prediction. By applying semiempirical finite-size scaling to the relevant data, the critical points obtained by the NN approach agree well with the known results established in the literature. This implies that for each of the considered models, the determination of its various phases requires only a little information. The outcomes presented here demonstrate convincingly that the employed universal NN is not only valid for the symmetry breaking related phase transitions, but also works for calculating the critical points of the phase transitions associated with topology. The efficiency of the used NN in the computation is examined by carrying out several detailed benchmark calculations.
The Berezinskii-Kosterlitz-Thouless transition is a very specific phase transition where all thermodynamic quantities are smooth.Therefore, it is difficult to determine the critical temperature in a precise way.In this paper we demonstrate how neural networks can be used to perform this task.In particular, we study how the accuracy of the transition identification depends on the way the neural networks are trained.We apply our approach to three different systems: (i) the classical XY model, (ii) the phase-fermion model, where classical and quantum degrees of freedom are coupled and (iii) the quantum XY model.
Abstract Machine learning (ML) has become a useful tool for studying phase transitions in statistical systems. For the two-dimensional classical XY model, however, the topological character of the Berezinskii–Kosterlitz–Thouless (BKT) transition and pronounced finite-size effects make it nontrivial to extract reproducible size-dependent pseudo-critical temperatures from configuration data. Existing studies often stop at phase classification, leaving open how standard neural-network outputs can be turned into quantitatively testable observables. Here we develop a ML-assisted framework for the 2D XY model that uses standard network outputs to extract a size-dependent sequence of operational pseudo-critical temperatures T ∗ ( L ) . Specifically, we generate Monte Carlo configurations using embedded cluster updates, train a standard ResNet18 only on samples from the Quasi-ordered Phase and the Disordered Phase, and determine T ∗ ( L ) from bootstrap-averaged probability curves using the 50% crossing criterion. We then analyze the finite-size drift of this temperature sequence using BKT-motivated scaling and compare it with susceptibility-peak temperatures. The selected T ∗ ( L ) values display an overall downward finite-size drift compatible with BKT-motivated behavior and lie within the conventional susceptibility fluctuation regime. We further characterize and assess the extracted scale through threshold-sensitivity tests, window-variation analysis, probability calibration, comparison with a simple-physics baseline, vortex-density measurement, input-encoding comparison, and saliency mapping. These diagnostics reveal that while the network response is dominated by macroscopic physical observables, the extracted T ∗ ( L ) is conditional on representation, augmentation, and model-selection protocols. These results demonstrate that standard classifier outputs can be converted into an operational finite-size temperature scale on a well-characterized benchmark model, while also revealing the dependence of that scale on the adopted inference protocol.