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arXiv:2005.00721 · Journal of Physics Condensed Matter

Learning what a machine learns in a many-body localization transition

Rubah Kausar, Wen-Jia Rao, Xin Wan

Published Jun 22, 2020• 7 citations
arXivHTMLPDF

Abstract

We employ a convolutional neural network to explore the distinct phases in random spin systems with the aim to understand the specific features that the neural network chooses to identify the phases. With the energy spectrum normalized to the bandwidth as the input data, we demonstrate that a network of the smallest nontrivial kernel width selects level spacing as the signature to distinguish the many-body localized phase from the thermal phase. We also study the performance of the neural network with an increased kernel width, based on which we find an alternative diagnostic to detect phases from the raw energy spectrum of such a disordered interacting system.

DOI
10.1088/1361-648x/ab9f09

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We employ a convolutional neural network to explore the distinct phases in random spin systems with the aim to understand the specific features that the neural network chooses to identify the phases. With the energy spectrum normalized to the bandwidth as the input data, we demonstrate that a network of the smallest nontrivial kernel width selects level spacing as the signature to distinguish the many-body localized phase from the thermal phase. We also study the performance of the neural network with an increased kernel width, based on which we find an alternative diagnostic to detect phases from the raw energy spectrum of such a disordered interacting system.
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.
The transition between thermal and many-body localized phases in isolated random spin systems are typically identified by the distribution of nearest level spacings. In this work, by employing machine learning methodology, we show this transition can be learnt through raw energy spectrum without any pre-processing. After achieving so in conventional random spin chain with differentiable level spacing distributions, we further construct novel models with misleading signatures of level spacing, and show machine can defeat the latter when training data is raw energy spectrum. Our work shows the low-level energy spectrum contains more information than level spacings, and can be captured by machine in a direct while efficient way, which makes it a promising new tool for studying a variety of isolated quantum systems.
We adapt a machine-learning approach to study the many-body localization transition in interacting fermionic systems on disordered one-dimensional (1D) and two-dimensional (2D) lattices. We perform supervised training of convolutional neural networks (CNNs) using labeled many-body wave functions at weak and strong disorder. In these limits, the average validation accuracy of the trained CNNs exceeds 99.95%. We use the disorder-averaged predictions of the CNNs to generate energy-resolved phase diagrams, which exhibit many-body mobility edges. We provide finite-size estimates of the critical disorder strengths at ${W}_{c}\ensuremath{\sim}2.8$ and 9.8 for 1D and 2D systems of 16 sites, respectively. Our results agree with the analysis of energy-level statistics and inverse participation ratio. By examining the convolutional layer, we unveil its feature extraction mechanism which highlights the pronounced peaks in localized many-body wave functions while rendering delocalized wave functions nearly featureless.
We study the many-body localization (MBL) transition in interacting fermionic systems on disordered one-dimensional lattices using a physics-informed machine-learning framework. Instead of feeding full many-body wave functions into the model, we construct a compact feature representation based on four physically motivated observables: the inverse participation ratio, the Shannon entropy, the many-body hybridization parameter, and the mean level-spacing ratio. These quantities capture complementary aspects of localization, entanglement, and spectral correlations, and are used to train a Kolmogorov--Arnold Network (KAN) classifier on eigenstates deep in the weak and strong disorder regimes. The resulting KAN achieves a validation accuracy exceeding 99.9%99.9\%99.9%, comparable to that of convolutional neural networks trained directly on high-dimensional wave-function data, while requiring substantially reduced input dimensionality and significantly shorter training time. Applying the trained classifier across the full energy spectrum yields energy-resolved phase diagrams that reveal a clear many-body mobility edge and provide a consistent estimate of the critical disorder strength. The approach is inherently extensible: additional physically relevant observables can be incorporated into the feature space in a systematic manner without altering the overall architecture. Our results demonstrate that feature-based learning with KAN provides an efficient, scalable, and interpretable methodology for identifying many-body localization transitions, offering a practical alternative to raw-data-based neural network approaches.
We show that a simple artificial neural network trained on entanglement spectra of individual states of a many-body quantum system can be used to determine the transition between a many-body localized and a thermalizing regime. Specifically, we study the Heisenberg spin-1/2 chain in a random external field. We employ a multilayer perceptron with a single hidden layer, which is trained on labeled entanglement spectra pertaining to the fully localized and fully thermal regimes. We then apply this network to classify spectra belonging to states in the transition region. For training, we use a cost function that contains, in addition to the usual error and regularization parts, a term that favors a confident classification of the transition region states. The resulting phase diagram is in good agreement with the one obtained by more conventional methods and can be computed for small systems. In particular, the neural network outperforms conventional methods in classifying individual eigenstates pertaining to a single disorder realization. It allows us to map out the structure of these eigenstates across the transition with spatial resolution. Furthermore, we analyze the network operation using the dreaming technique to show that the neural network correctly learns by itself the power-law structure of the entanglement spectra in the many-body localized regime.
Determining phase diagrams and phase transitions semiautomatically using machine learning has received a lot of attention recently, with results in good agreement with more conventional approaches in most cases. When it comes to more quantitative predictions, such as the identification of universality class or precise determination of critical points, the task is more challenging. As an exacting testbed, we study the Heisenberg spin-1/2 chain in a random external field that is known to display a transition from a many-body localized to a thermalizing regime, which nature is not entirely characterized. We introduce different neural network structures and dataset setups to achieve a finite-size scaling analysis with the least possible physical bias (no assumed knowledge on the phase transition and directly inputting wave-function coefficients), using state-of-the-art input data simulating chains of sizes up to L=24L=24L=24. In particular, we use domain adversarial techniques to ensure that the network learns scale-invariant features. We find a variability of the output results with respect to network and training parameters, resulting in relatively large uncertainties on final estimates of critical point and correlation length exponent which tend to be larger than the values obtained from conventional approaches. We put the emphasis on interpretability throughout the paper and discuss what the network appears to learn for the various used architectures. Our findings show that a quantitative analysis of phase transitions of unknown nature remains a difficult task with neural networks when using the minimally engineered physical input.