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arXiv:1809.10419 · Physical review. B./Physical review. B

Self-organizing maps as a method for detecting phase transitions and phase identification

Albert A. Shirinyan, Valerii K. Kozin, Johan Hellsvik, Manuel Pereiro, Olle Eriksson, Dmitry I. Yudin

Published Jan 10, 2019• 25 citations
arXivHTMLPDF

Abstract

Originating from image recognition, methods of machine learning allow for effective feature extraction and dimensionality reduction in multidimensional datasets, thereby providing an extraordinary tool to deal with classical and quantum models in many-body physics. In this study, we employ a specific unsupervised machine learning technique---self-organizing maps---to create a low-dimensional representation of microscopic states, relevant for macroscopic phase identification and detecting phase transitions. We explore the properties of spin Hamiltonians of two archetype model systems: a two-dimensional Heisenberg ferromagnet and a three-dimensional crystal, Fe in the body-centered-cubic structure. The method of self-organizing maps, which is known to conserve connectivity of the initial dataset, is compared to the cumulant method theory and is shown to be as accurate while being computationally more efficient in determining a phase transition temperature. We argue that the method proposed here can be applied to explore a broad class of second-order phase-transition systems, not only magnetic systems but also, for example, order-disorder transitions in alloys.

DOI
10.1103/physrevb.99.041108

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•Mar 8, 2019•6 citations
Originating from image recognition, methods of machine learning allow for effective feature extraction and dimensionality reduction in multidimensional datasets, thereby providing an extraordinary tool to deal with classical and quantum models in many-body physics. In this study, we employ a specific unsupervised machine learning technique---self-organizing maps---to create a low-dimensional representation of microscopic states, relevant for macroscopic phase identification and detecting phase transitions. We explore the properties of spin Hamiltonians of two archetype model systems: a two-dimensional Heisenberg ferromagnet and a three-dimensional crystal, Fe in the body-centered-cubic structure. The method of self-organizing maps, which is known to conserve connectivity of the initial dataset, is compared to the cumulant method theory and is shown to be as accurate while being computationally more efficient in determining a phase transition temperature. We argue that the method proposed here can be applied to explore a broad class of second-order phase-transition systems, not only magnetic systems but also, for example, order-disorder transitions in alloys.
The classification of phase transitions is a central and challenging task in condensed matter physics. Typically, it relies on the identification of order parameters and the analysis of singularities in the free energy and its derivatives. Here, we propose an alternative framework to identify quantum phase transitions, employing both unsupervised and supervised machine learning techniques. Using the axial next-nearest-neighbor Ising (ANNNI) model as a benchmark, we show how unsupervised learning can detect three phases (ferromagnetic, paramagnetic, and a cluster of the antiphase with the floating phase) as well as two distinct regions within the paramagnetic phase. Employing supervised learning we show that transfer learning becomes possible: a machine trained only with nearest-neighbor interactions can learn to identify a new type of phase occurring when next-nearest-neighbor interactions are introduced. All our results rely on few- and low-dimensional input data (up to twelve lattice sites), thus providing a computational friendly and general framework for the study of phase transitions in many-body systems.
Unsupervised learning is a discipline of machine learning which aims at discovering patterns in large data sets or classifying the data into several categories without being trained explicitly. We show that unsupervised learning techniques can be readily used to identify phases and phases transitions of many-body systems. Starting with raw spin configurations of a prototypical Ising model, we use principal component analysis to extract relevant low-dimensional representations of the original data and use clustering analysis to identify distinct phases in the feature space. This approach successfully finds physical concepts such as the order parameter and structure factor to be indicators of a phase transition. We discuss the future prospects of discovering more complex phases and phase transitions using unsupervised learning techniques.
We develop a self-supervised ensemble learning (SSEL) method to accurately classify distinct types of phase transitions by analyzing the fluctuation properties of machine learning outputs. Employing the 2D Potts model and the 2D Clock model as benchmarks, we demonstrate the capability of SSEL in discerning first-order, second-order, and Berezinskii-Kosterlitz-Thouless transitions, using in situ spin configurations as the input features. Furthermore, we show that the SSEL approach can also be applied to investigate quantum phase transitions in 1D Ising and 1D XXZXXZXXZ models upon incorporating quantum sampling. We argue that the SSEL model simulates a special state function with higher-order correlations between physical quantities, and hence provides richer information than previous machine learning methods. Consequently, our SSEL method can be generally applied to the identification/classification of phase transitions even without explicit knowledge of the underlying theoretical models.
The application of state-of-the-art machine learning techniques to statistical physic problems has seen a surge of interest for their ability to discriminate phases of matter by extracting essential features in the many-body wavefunction or the ensemble of correlators sampled in Monte Carlo simulations. Here we introduce a gener- alization of supervised machine learning approaches that allows to accurately map out phase diagrams of inter- acting many-body systems without any prior knowledge, e.g. of their general topology or the number of distinct phases. To substantiate the versatility of this approach, which combines convolutional neural networks with quantum Monte Carlo sampling, we map out the phase diagrams of interacting boson and fermion models both at zero and finite temperatures and show that first-order, second-order, and Kosterlitz-Thouless phase transitions can all be identified. We explicitly demonstrate that our approach is capable of identifying the phase transition to non-trivial many-body phases such as superfluids or topologically ordered phases without supervision.
Using machine learning (ML) to recognize different phases of matter and to infer the entire phase diagram has proven to be an effective tool given a large dataset. In our previous proposals, we have successfully explored phase transitions for topological phases of matter at low dimensions either in a supervised or an unsupervised learning protocol with the assistance of quantum-information-related quantities. In this work, we adopt our previous ML procedures to study quantum phase transitions of magnetism systems such as the XYXYXY and XXZXXZXXZ spin chains by using spin-spin correlation functions as the input data. We find that our proposed approach not only maps out the phase diagrams with accurate phase boundaries, but also indicates some features that have not been observed in the field of machine learning before. In particular, we define so-called relevant correlation functions to some corresponding phases that can always distinguish between those and their neighbors. Based on the unsupervised learning protocol we proposed [Phys. Rev. B 104, 165108 (2021)], the reduced latent representations of the inputs combined with the clustering algorithm show the connectedness or disconnectedness between neighboring clusters (phases) just corresponding to the continuous or disrupt quantum phase transition, respectively. This property reminds us of the behavior of order parameters. Moreover, in the silhouette analysis we show that the ferromagnetic states in the XXZXXZXXZ model with various anisotropy parameters correspond to almost the same silhouette value, while the critical or antiferromagnetic states behave quite differently. The analysis further indicates that the minima of silhouette values are close to the phase-transition points, showing strong positive correlation. These results again justify the usefulness of our proposed ML procedures, and they move us a step further toward understanding the relation between ML and quantum phase transitions from correlation function aspects.
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.