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arXiv:2010.04730 · Physical Review Research

Interpretable and unsupervised phase classification

Julian Arnold, Frank Schäfer, Martin Žonda, Axel U. J. Lode

Published Jul 15, 2021• 34 citations
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Abstract

Fully automated classification methods that provide direct physical insights into phase diagrams are of current interest. Interpretable, i.e., fully explainable, methods are desired for which we understand why they yield a given phase classification. Ideally, phase classification methods should also be unsupervised. That is, they should not require prior labeling or knowledge of the phases of matter to be characterized. Here, we demonstrate an unsupervised machine-learning method for phase classification, which is rendered interpretable via an analytical derivation of the functional relationship between its optimal predictions and the input data. Based on these findings, we propose and apply an alternative, physically-motivated, data-driven scheme, which relies on the difference between mean input features. This mean-based method does not rely on any predictive model and is thus computationally cheap and directly explainable. As an example, we consider the physically rich ground-state phase diagram of the spinless Falicov-Kimball model.

DOI
10.1103/physrevresearch.3.033052

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Fully automated classification methods that provide direct physical insights into phase diagrams are of current interest. Interpretable, i.e., fully explainable, methods are desired for which we understand why they yield a given phase classification. Ideally, phase classification methods should also be unsupervised. That is, they should not require prior labeling or knowledge of the phases of matter to be characterized. Here, we demonstrate an unsupervised machine-learning method for phase classification, which is rendered interpretable via an analytical derivation of the functional relationship between its optimal predictions and the input data. Based on these findings, we propose and apply an alternative, physically-motivated, data-driven scheme, which relies on the difference between mean input features. This mean-based method does not rely on any predictive model and is thus computationally cheap and directly explainable. As an example, we consider the physically rich ground-state phase diagram of the spinless Falicov-Kimball model.
We apply various unsupervised machine learning methods for phase classification to investigate the finite-temperature phase diagram of the spinless Falicov-Kimball model in two dimensions. Using only particle occupation snapshots from Monte Carlo simulations as input, each technique, including a straightforward classification based on principal component analysis (PCA), successfully identifies the phase boundary between ordered and disordered phases, independent of the type of phase transition. Remarkably, these techniques also distinguish between the weakly localized and Anderson-localized regimes within the disordered phase, accurately identifying their crossover, which is a challenging task for standard methods. Among the machine learning approaches used, PCA based analysis outperforms more complex methods, such as neural network predictors and autoencoders. These results underscore the effectiveness of simple unsupervised techniques in examining phase transitions and electron localization in complex correlated systems.
Phase diagrams in condensed-matter physics are often very complex. The development of methods that can identify distinct phases without any prior knowledge is, therefore, of great interest. In this study, we applied unsupervised machine learning methods to find phase boundaries in the Falicov-Kimball model, using principal component analysis and the prediction-based method. We showed that both methods can distinguish the ordered from the disordered phase. Moreover, these methods are able to distinguish the weakly localized phase from the Anderson insulator phase, which both exist within the disordered phase. Keywords: prediction-based method, principal component analysis, Falicov-Kimball model, unsupervised phase classification, Machine Learning
Machine learning techniques have been shown to be effective to recognize different phases of matter and produce phase diagrams in the parameter space of interest, while they usually require prior labeled data to perform well. Here, we propose a machine learning procedure, mainly in an unsupervised manner, which can first identify topological/nontopological phases and then refine the locations of phase boundaries. By following this proposed procedure, we expand our previous work on the one-dimensional ppp-wave superconductor [Tsai et al., Phys. Rev. B 102, 054512 (2020)] and further on the Su-Schrieffer-Heeger model, with an emphasis on using the quantum-entanglement-based quantities as the input features. We find that our method not only reproduces similar results to the previous work with sharp phase boundaries but importantly it also does not rely on prior knowledge of the phase space, e.g., the number of phases present. We conclude with a few remarks about its potential, limitations, and explanatory power.
We demonstrate how to explore phase diagrams with automated and unsupervised machine learning to find regions of interest for possible new phases. In contrast to supervised learning, where data is classified using predetermined labels, we here perform anomaly detection, where the task is to differentiate a normal dataset, composed of one or several classes, from anomalous data. As a paradigmatic example, we explore the phase diagram of the extended Bose Hubbard model in one dimension at exact integer filling and employ deep neural networks to determine the entire phase diagram in a completely unsupervised and automated fashion. As input data for learning, we first use the entanglement spectra and central tensors derived from tensor-networks algorithms for ground-state computation and later we extend our method and use experimentally accessible data such as low-order correlation functions as inputs. Our method allows us to reveal a phase-separated region between supersolid and superfluid parts with unexpected properties, which appears in the system in addition to the standard superfluid, Mott insulator, Haldane-insulating, and density wave phases.
Identifying phase transitions and classifying phases of matter is central to understanding the properties and behavior of a broad range of material systems. In recent years, machine-learning (ML) techniques have been successfully applied to perform such tasks in a data-driven manner. However, the success of this approach notwithstanding, we still lack a clear understanding of ML methods for detecting phase transitions, particularly of those that utilize neural networks (NNs). In this work, we derive analytical expressions for the optimal output of three widely used NN-based methods for detecting phase transitions. These optimal predictions correspond to the results obtained in the limit of high model capacity. Therefore, in practice they can, for example, be recovered using sufficiently large, well-trained NNs. The inner workings of the considered methods are revealed through the explicit dependence of the optimal output on the input data. By evaluating the analytical expressions, we can identify phase transitions directly from experimentally accessible data without training NNs, which makes this procedure favorable in terms of computation time. Our theoretical results are supported by extensive numerical simulations covering, e.g., topological, quantum, and many-body localization phase transitions. We expect similar analyses to provide a deeper understanding of other classification tasks in condensed matter physics.
Machine-learning driven models have proven to be powerful tools for the identification of phases of matter. In particular, unsupervised methods hold the promise to help discover new phases of matter without the need for any prior theoretical knowledge. While for phases characterized by a broken symmetry, the use of unsupervised methods has proven to be successful, topological phases without a local order parameter seem to be much harder to identify without supervision. Here, we use an unsupervised approach to identify boundaries of the topological phases. We train artificial neural nets to relate configurational data or measurement outcomes to quantities like temperature or tuning parameters in the Hamiltonian. The accuracy of these predictive models can then serve as an indicator for phase transitions. We successfully illustrate this approach on both the classical Ising gauge theory as well as on the quantum ground state of a generalized toric code.