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Unsupervised identification of the phase transition on the 2D-Ising model

Constantia Alexandrou, Andreas Athenodorou, Charalambos Chrysostomou, Srijit Paul

Published Mar 8, 2019• 6 citations
OpenAlex

Abstract

We investigate deep learning autoencoders for the unsupervised recognition of phase transitions in physical systems formulated on a lattice. We use spin configurations produced for the 2-dimensional ferromagnetic Ising model in zero external magnetic field. We study numerically the relation between one latent dimension extracted from the autoencoder to the critical temperature $T_c$. The autoencoder reveals the two phases, one for which the spins are ordered and the other for which spins are disordered, reflecting the restoration of the $\mathbb{Z}_2$ symmetry as the temperature increases. For the largest volume studied, the transition between the two phases occurs very close to the theoretically extracted critical temperature. We define as a quasi-order parameter the absolute average latent dimension ${\tilde z}$, which enables us to predict the critical temperature. We show that one can build the latent susceptibility and use it to quantify the value of the critical temperature $T_c(L)$ at different lattice sizes and that these values suffer from only small finite scaling effects. We demonstrate that $T_c(L)$ extrapolates to the known theoretical value as $L \to \infty$ suggesting that the autoencoder can also be used to extract the critical temperature of the phase transition to an adequate precision.

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We investigate deep learning autoencoders for the unsupervised recognition of phase transitions in physical systems formulated on a lattice. We use spin configurations produced for the 2-dimensional ferromagnetic Ising model in zero external magnetic field. We study numerically the relation between one latent dimension extracted from the autoencoder to the critical temperature TcT_cTc​. The autoencoder reveals the two phases, one for which the spins are ordered and the other for which spins are disordered, reflecting the restoration of the Z2\mathbb{Z}_2Z2​ symmetry as the temperature increases. For the largest volume studied, the transition between the two phases occurs very close to the theoretically extracted critical temperature. We define as a quasi-order parameter the absolute average latent dimension z~{\tilde z}z~, which enables us to predict the critical temperature. We show that one can build the latent susceptibility and use it to quantify the value of the critical temperature Tc(L)T_c(L)Tc​(L) at different lattice sizes and that these values suffer from only small finite scaling effects. We demonstrate that Tc(L)T_c(L)Tc​(L) extrapolates to the known theoretical value as L→∞L \to \inftyL→∞ suggesting that the autoencoder can also be used to extract the critical temperature of the phase transition to an adequate precision.
Abstract We investigate deep learning autoencoders for the unsupervised recognition of phase transitions in physical systems formulated on a lattice. We focus our investigation on the 2-dimensional ferromagnetic Ising model and then test the application of the autoencoder on the anti-ferromagnetic Ising model. We use spin configurations produced for the 2-dimensional ferromagnetic and anti-ferromagnetic Ising model in zero external magnetic field. For the ferromagnetic Ising model, we study numerically the relation between one latent variable extracted from the autoencoder to the critical temperature T c . The proposed autoencoder reveals the two phases, one for which the spins are ordered and the other for which spins are disordered, reflecting the restoration of the ℤ 2 symmetry as the temperature increases. We provide a finite volume analysis for a sequence of increasing lattice sizes. For the largest volume studied, the transition between the two phases occurs very close to the theoretically extracted critical temperature. We define as a quasi-order parameter the absolute average latent variable z ̃ , which enables us to predict the critical temperature. One can define a latent susceptibility and use it to quantify the value of the critical temperature T c ( L ) at different lattice sizes and that these values suffer from only small finite scaling effects. We demonstrate that T c ( L ) extrapolates to the known theoretical value as L → ∞ suggesting that the autoencoder can also be used to extract the critical temperature of the phase transition to an adequate precision. Subsequently, we test the application of the autoencoder on the anti-ferromagnetic Ising model, demonstrating that the proposed network can detect the phase transition successfully in a similar way. Graphical abstract
We investigate the application of deep learning techniques employing the conditional variational autoencoders for semi-supervised learning of latent parameters to describe phase transition in the two-dimensional (2D) ferromagnetic Ising model and the two-dimensional XY model. For both models, we utilize spin configurations generated using the Wolff algorithms below and above the critical temperatures. For the 2D Ising model we find the latent parameter of conditional variational autoencoders is correlated to the known order parameter of magnetization more efficiently than their correspondence in variational autoencoders used previously. It can also clearly identify the restoration of the Z2\mathbb{Z}_2Z2​ symmetry beyond the critical point. The critical temperature extracted from the latent parameter at larger lattices are found to be approaching its correct value. Similarly, for the 2D XY model, we find our chosen network with the latent representation of conditional variational autoencoders is equally capable of separating the two phases between the high and low temperatures, again at the correct critical temperature with reasonable accuracy. Together these results show that the latent representation of conditional variational autoencoders can be employed efficiently to identify the phases of condensed matter systems, without their prior knowledge.
Machine learning methods have been important in the study of phase transitions. Unsupervised methods are particularly attractive because they do not require prior knowledge of the existence of a phase transition. In this work we focus on the constant magnetization Ising model in two (2D) and three (3D) dimensions. While there have been many studies using machine learning for the critical behavior of these systems, we are not aware of any studies for the phase diagram at off-critical magnetizations below the critical temperature. Previous work has used the raw spins as the input feature. We show that a more robust input feature is the local affinity, where the value of the feature at each site is determined by the spin and its neighbors. When coupled with a variational autoencoder, the method is able to predict the phase behavior of the 2D and 3D Ising models (including the critical exponent β) in quantitative agreement with conventional simulations. The choice of activation functions in the autoencoder is crucial, and this requires physical insight into the nature of the phase transition. The method is general and can be applied to any lattice or off-lattice system.
We develop a one-class, deep-learning framework to detect the phase transition and recover critical behavior of the three-dimensional (3D) Ising model. A 3D convolutional autoencoder (CAE) is trained on ground-state configurations only, without prior knowledge of the critical temperature, the Hamiltonian, or the order parameter. After training, the model is applied to Monte Carlo configurations across a wide temperature range and different lattice sizes. The mean-square reconstruction error is shown to be sensitive to the transition. Finite-size scaling of the peak location for the reconstruction error susceptibility yields the critical temperature T c = 4.5128 ( 58 ) , in excellent agreement with the known value. Additionally, we obtain an estimate of the correlation-length critical exponent, ν = 0.63 ( 27 ) , also consistent with results from the literature. Our results show that a one-class CAE, trained on zero-temperature configurations only, can recover nontrivial critical behavior of the 3D Ising model.
The 2-dimensional Ising model on a square lattice is investigated with a variational autoencoder in the non-vanishing field case for the purpose of extracting the crossover region between the ferromagnetic and paramagnetic phases. The encoded latent variable space is found to provide suitable metrics for tracking the order and disorder in the Ising configurations that extends to the extraction of a crossover region in a way that is consistent with expectations. The extracted results achieve an exceptional prediction for the critical point as well as agreement with previously published results on the configurational magnetizations of the model. The performance of this method provides encouragement for the use of machine learning to extract meaningful structural information from complex physical systems where little a priori data is available.
The 2-dimensional Ising model on a square lattice is investigated with a variational autoencoder in the non-vanishing field case for the purpose of extracting the crossover region between the ferromagnetic and paramagnetic phases. The encoded latent variable space is found to provide suitable metrics for tracking the order and disorder in the Ising configurations that extends to the extraction of a crossover region in a way that is consistent with expectations. The extracted results achieve an exceptional prediction for the critical point as well as agreement with previously published results on the configurational magnetizations of the model. The performance of this method provides encouragement for the use of machine learning to extract meaningful structural information from complex physical systems where little a priori data is available.