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arXiv:2108.05737

Neural Networks as Universal Probes of Many-Body Localization in Quantum Graphs

Cameron Beetar, Jeff Murugan, Dario Rosa

Published Aug 12, 2021• 1 citation
arXivHTMLPDF

Abstract

We show that a neural network, trained on the entanglement spectra of a nearest neighbor Heisenberg chain in a random transverse magnetic field, can be used to efficiently study the ergodic/many-body localized properties of a number of other quantum systems, without further re-training. We benchmark our computational architecture against a $J_{1}\!\!-\!\!J_{2}$ - model, which extends the Heisenberg chain to include next-to-nearest neighbor interactions, with excellent agreement with known results. When applied to Hamiltonians that differ from the training model by the topology of the underlying graph, the neural network is able to predict the critical disorders and, more generally, shapes of the mobility edge that agree with our heuristic expectation. We take this as proof of principle that machine learning algorithms furnish a powerful new set of computational tools to explore the many-body physics frontier.

DOI
10.48550/arxiv.2108.05737

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We show that a neural network, trained on the entanglement spectra of a nearest neighbor Heisenberg chain in a random transverse magnetic field, can be used to efficiently study the ergodic/many-body localized properties of a number of other quantum systems, without further re-training. We benchmark our computational architecture against a J1 ⁣ ⁣− ⁣ ⁣J2J_{1}\!\!-\!\!J_{2}J1​−J2​ - model, which extends the Heisenberg chain to include next-to-nearest neighbor interactions, with excellent agreement with known results. When applied to Hamiltonians that differ from the training model by the topology of the underlying graph, the neural network is able to predict the critical disorders and, more generally, shapes of the mobility edge that agree with our heuristic expectation. We take this as proof of principle that machine learning algorithms furnish a powerful new set of computational tools to explore the many-body physics frontier.
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.
The numerical emulation of quantum systems often requires an exponential number of degrees of freedom which translates to a computational bottleneck. Methods of machine learning have been used in adjacent fields for effective feature extraction and dimensionality reduction of high-dimensional datasets. Recent studies have revealed that neural networks are further suitable for the determination of macroscopic phases of matter and associated phase transitions as well as efficient quantum state representation. In this work, we address quantum phase transitions in quantum spin chains, namely the transverse field Ising chain and the anisotropic XY chain, and show that even neural networks with no hidden layers can be effectively trained to distinguish between magnetically ordered and disordered phases. Our neural network acts to predict the corresponding crossovers finite-size systems undergo. Our results extend to a wide class of interacting quantum many-body systems and illustrate the wide applicability of neural networks to many-body quantum physics.
Abstract The numerical emulation of quantum systems often requires an exponential number of degrees of freedom which translates to a computational bottleneck. Methods of machine learning have been used in adjacent fields for effective feature extraction and dimensionality reduction of high-dimensional datasets. Recent studies have revealed that neural networks are further suitable for the determination of macroscopic phases of matter and associated phase transitions as well as efficient quantum state representation. In this work, we address quantum phase transitions in quantum spin chains, namely the transverse field Ising chain and the anisotropic XY chain, and show that even neural networks with no hidden layers can be effectively trained to distinguish between magnetically ordered and disordered phases. Our neural network acts to predict the corresponding crossovers finite-size systems undergo. Our results extend to a wide class of interacting quantum many-body systems and illustrate the wide applicability of neural networks to many-body quantum physics.
The study of interacting quantum many-body systems is constrained by the exponential growth of the Hilbert space with system size, which limits both numerical simulations and theoretical understanding. This thesis explores how the interplay between disorder and interactions shapes localization and ergodicity in quantum many-body systems, using the exact diagonalization method, and how machine learning can serve as an alternative approach for studying these systems. We first investigate the many-body localization problem in a one-dimensional SU(2)-invariant random antiferromagnetic Heisenberg chain using exact and sparse matrix diagonalization methods. By analyzing energy level gap-ratio statistics, participation ratio, entanglement entropy and the entanglement spectral parameter as a function of disorder strength under di!erent random coupling distributions, we identify clear qualitative di!erences from the widely studied random field Heisenberg chain. For system sizes up to L = 24, signatures of localized behaviour are much less pronounced than in the case of the random field model. Motivated by the limitations of conventional numerical methods, we examine machine learning as a scalable paradigm for many-body physics. Using the Bose- Hubbard model in one and two dimensions as a benchmark, we test HUBBARDNET, a deep neural network architecture, by replacing standard stochastic gradient descent with adaptive moment estimation (ADAM), adjusting the learning rate for improved optimization, and introducing physics-informed output activations that can resolve extremely small wave-function amplitudes. This approach yields ground-state energy errors reduced by orders of magnitude and wave function fidelities exceeding 99%, remaining robust across four decades of interaction strength and over superfluid, Mott-insulating, and quantum chaotic regimes. Beyond energy prediction, the trained models accurately reproduce generalized inverse participation ratios and multifractal dimensions for ground and excited states, capturing localization, delocalization, and multifractality across the spectrum, with excited-state training explicitly guided by fractal dimensions that encode the multifractal properties of the wave functions.
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.
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.