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arXiv:1711.00020 · Physical Review Letters

Machine Learning Out-of-Equilibrium Phases of Matter

Jordan Venderley, Vedika Khemani, Eun-Ah Kim

Published Jun 21, 2018• 141 citations
arXivHTMLPDF

Abstract

Neural-network-based machine learning is emerging as a powerful tool for obtaining phase diagrams when traditional regression schemes using local equilibrium order parameters are not available, as in many-body localized (MBL) or topological phases. Nevertheless, instances of machine learning offering new insights have been rare up to now. Here we show that a single feed-forward neural network can decode the defining structures of two distinct MBL phases and a thermalizing phase, using entanglement spectra obtained from individual eigenstates. For this, we introduce a simplicial geometry-based method for extracting multipartite phase boundaries. We find that this method outperforms conventional metrics for identifying MBL phase transitions, revealing a sharper phase boundary and shedding new insight on the topology of the phase diagram. Furthermore, the phase diagram we acquire from a single disorder configuration confirms that the machine-learning-based approach we establish here can enable speedy exploration of large phase spaces that can assist with the discovery of new MBL phases. To our knowledge, this Letter represents the first example of a standard machine learning approach revealing new information on phase transitions.

DOI
10.1103/physrevlett.120.257204

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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.
We identify a new ``order parameter'' for the disorder-driven many-body localization transition by leveraging machine learning. Contrary to previous studies, our method is almost entirely unsupervised. A game theoretic process between neural networks defines an adversarial setup with conflicting objectives to identify what characteristic features to base efficient predictions on. This reduces the numerical effort for mapping out the phase diagram by a factor of $100\ifmmode\times\else\texttimes\fi{}$ and allows us to pin down the transition, as the point at which the physics changes qualitatively, in an objective and cleaner way than is possible with the existing diverse array of quantities. Our approach of automated discovery is applicable specifically to poorly understood phase transitions and is a starting point for a research program leveraging the potential of machine learning--assisted research in physics.
We employ the methods of machine learning to study the many-body localization (MBL) transition in a 1D random spin system. By using the raw energy spectrum without pre-processing as training data, it is shown that the MBL transition point is correctly predicted by the machine. The structure of the neural network reveals the nature of this dynamical phase transition that involves all energy levels, while the bandwidth of the spectrum and nearest level spacing are the two dominant patterns and the latter stands out to classify phases. We further use a comparative unsupervised learning method, i.e., principal component analysis, to confirm these results.
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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.
We propose the use of recurrent neural networks for classifying phases of matter based on the dynamics of experimentally accessible observables. We demonstrate this approach by training recurrent networks on the magnetization traces of two distinct models of one-dimensional disordered and interacting spin chains. The obtained phase diagram for a well-studied model of the many-body localization transition shows excellent agreement with previously known results obtained from time-independent entanglement spectra. For a periodically driven model featuring an inherently dynamical time-crystalline phase, the phase diagram that our network traces coincides with an order parameter for its expected phases.