Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2108.04244 · Physical review. B./Physical review. B

Distinguishing an Anderson insulator from a many-body localized phase through space-time snapshots with neural networks

Florian Kotthoff, Frank Pollmann, Giuseppe De Tomasi

Published Dec 27, 2021• 15 citations
arXivHTMLPDF

Abstract

Distinguishing the dynamics of an Anderson insulator from a many-body localized (MBL) phase is an experimentally challenging task. In this work we propose a method based on machine learning techniques to analyze experimental snapshot data to separate the two phases. We show how to train three-dimensional convolutional neural networks (CNNs) using space-time Fock-state snapshots, allowing us to obtain dynamic information about the system. We benchmark our method on a paradigmatic model showing MBL ($t\text{\ensuremath{-}}V$ model with quenched disorder), where we obtain a classification accuracy of $\ensuremath{\approx}80%$ between an Anderson insulator and an MBL phase. We underline the importance of providing temporal information to the CNNs and we show that CNNs learn the crucial difference between an Anderson localized and an MBL phase, namely the difference in the propagation of quantum correlations. Particularly, we show that the misclassified MBL samples are characterized by an unusually slow propagation of quantum correlations, and thus the CNNs label them wrongly as Anderson localized. Finally, we apply our method to the case with quasiperiodic potential, known as the Aubry-Andr\'e model (AA model). We find that the CNNs have more difficulties in separating the two phases. We show that these difficulties are due to the fact that the MBL phase of the AA model is characterized by a slower information propagation for numerically accessible system sizes.

DOI
10.1103/physrevb.104.224307

Related papers

Analyzing Nonequilibrium Quantum States through Snapshots with Artificial Neural Networks

Annabelle Bohrdt, Sooshin Kim, Alexander Lukin +5 more

Current quantum simulation experiments are starting to explore nonequilibrium many-body dynamics in previously inaccessible regimes in terms of system sizes and timescales. Therefore, the question emerges as to which observables are best suited to study the dynamics in such quantum many-body systems. Using machine learning techniques, we investigate the dynamics and, in particular, the thermalization behavior of an interacting quantum system that undergoes a nonequilibrium phase transition from an ergodic to a many-body localized phase. We employ supervised and unsupervised training methods to distinguish nonequilibrium from equilibrium data, using the network performance as a probe for the thermalization behavior of the system. We test our methods with experimental snapshots of ultracold atoms taken with a quantum gas microscope. Our results provide a path to analyze highly entangled large-scale quantum states for system sizes where numerical calculations of conventional observables become challenging.

arXiv:2012.11586•Physical Review Letters•Oct 7, 2021•30 citations

Supervised and unsupervised learning of the many-body critical phase, phase transitions, and critical exponents in disordered quantum systems

Aamna Ahmed, Nilanjan Roy

Supervised and unsupervised machine-learning-based frameworks have been implemented in the past to classify various dynamical phases of disordered quantum systems using rather sophisticated (highly preprocessed or difficult to generate) input data. In this paper, we begin by questioning the existence of a nonergodic extended phase, namely, the many-body critical (MBC) phase in finite systems of an interacting quasiperiodic system. We find that this phase can be separately detected from the other phases, such as the many-body ergodic (ME) and many-body localized (MBL) phases, in the model through supervised neural networks made for both binary and multiclass classification tasks, utilizing, rather unpreprocessed, eigenvalue spacings and eigenvector probability densities as input features. Moreover, the output of our trained neural networks can also indicate the critical points separating ME, MBC, and MBL phases, which are consistent with the same obtained from other conventional methods. We also employ unsupervised learning techniques, particularly principal component analysis (PCA) of eigenvector probability densities, to investigate how this framework, without any training, captures the, rather unknown, many-body phases (ME, MBL, and MBC) and single-particle phases (delocalized, localized, and critical) of the interacting and noninteracting systems, respectively. Our findings reveal that PCA entropy serves as an effective indicator (order parameter) for detecting phase transitions in the single-particle systems. Moreover, this method proves applicable to many-body systems when the data undergoes a suitable preprocessing. Interestingly, when it comes to the extraction of critical (correlation length) exponents through a finite-size scaling, we find that for single-particle systems, scaling collapse of neural network outputs is obtained using components of inverse participation ratio (IPR) as input data. Remarkably, we observe identical critical exponents as obtained from the scaling collapse of the IPR directly for different single-particle phase transitions. However, the current finite-size scaling approach to the network output cannot reliably extract critical exponents for the many-body system that needs further investigation.

arXiv:2501.03981•Physical review. B./Physical review. B•May 22, 2025•5 citations

Unsupervised machine learning phase classification for the Falicov-Kimball model

Lukáš Frk, Pavel Baláž, Elguja Archemashvili +1 more

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.

arXiv:2411.07319•Physical review. B./Physical review. B•May 13, 2025•3 citations

Many-body mobility edges in one and two dimensions revealed by convolutional neural networks

Anffany Chen

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.

arXiv:2312.09223•Physical review. B./Physical review. B•Feb 12, 2024•2 citations

Many-body mobility edges in one dimension revealed by efficient and interpretable feature-based learning with Kolmogorov-Arnold Networks

Siqi Dai, Tian-Cheng Yi, Xingbo Wei +1 more

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 $99.9\%$, 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.

arXiv:2603.21807•Mar 23, 2026•0 citations

Quantum Mutual Information as a Probe for Many-Body Localization

Giuseppe De Tomasi, Soumya Bera, Jens Hjorleifur Bardarson +1 more

We demonstrate that the quantum mutual information (QMI) is a useful probe to study many-body localization (MBL). First, we focus on the detection of a metal-insulator transition for two different models, the noninteracting Aubry-André-Harper model and the spinless fermionic disordered Hubbard chain. We find that the QMI in the localized phase decays exponentially with the distance between the regions traced out, allowing us to define a correlation length, which converges to the localization length in the case of one particle. Second, we show how the QMI can be used as a dynamical indicator to distinguish an Anderson insulator phase from a MBL phase. By studying the spread of the QMI after a global quench from a random product state, we show that the QMI does not spread in the Anderson insulator phase but grows logarithmically in time in the MBL phase.

arXiv:1608.07183•Physical Review Letters•Jan 5, 2017•104 citations
Distinguishing the dynamics of an Anderson insulator from a many-body localized (MBL) phase is an experimentally challenging task. In this work we propose a method based on machine learning techniques to analyze experimental snapshot data to separate the two phases. We show how to train three-dimensional convolutional neural networks (CNNs) using space-time Fock-state snapshots, allowing us to obtain dynamic information about the system. We benchmark our method on a paradigmatic model showing MBL (t\ensuremath-Vt\text{\ensuremath{-}}Vt\ensuremath-V model with quenched disorder), where we obtain a classification accuracy of \ensuremath≈80\ensuremath{\approx}80%\ensuremath≈80 between an Anderson insulator and an MBL phase. We underline the importance of providing temporal information to the CNNs and we show that CNNs learn the crucial difference between an Anderson localized and an MBL phase, namely the difference in the propagation of quantum correlations. Particularly, we show that the misclassified MBL samples are characterized by an unusually slow propagation of quantum correlations, and thus the CNNs label them wrongly as Anderson localized. Finally, we apply our method to the case with quasiperiodic potential, known as the Aubry-Andr'e model (AA model). We find that the CNNs have more difficulties in separating the two phases. We show that these difficulties are due to the fact that the MBL phase of the AA model is characterized by a slower information propagation for numerically accessible system sizes.
Current quantum simulation experiments are starting to explore nonequilibrium many-body dynamics in previously inaccessible regimes in terms of system sizes and timescales. Therefore, the question emerges as to which observables are best suited to study the dynamics in such quantum many-body systems. Using machine learning techniques, we investigate the dynamics and, in particular, the thermalization behavior of an interacting quantum system that undergoes a nonequilibrium phase transition from an ergodic to a many-body localized phase. We employ supervised and unsupervised training methods to distinguish nonequilibrium from equilibrium data, using the network performance as a probe for the thermalization behavior of the system. We test our methods with experimental snapshots of ultracold atoms taken with a quantum gas microscope. Our results provide a path to analyze highly entangled large-scale quantum states for system sizes where numerical calculations of conventional observables become challenging.
Supervised and unsupervised machine-learning-based frameworks have been implemented in the past to classify various dynamical phases of disordered quantum systems using rather sophisticated (highly preprocessed or difficult to generate) input data. In this paper, we begin by questioning the existence of a nonergodic extended phase, namely, the many-body critical (MBC) phase in finite systems of an interacting quasiperiodic system. We find that this phase can be separately detected from the other phases, such as the many-body ergodic (ME) and many-body localized (MBL) phases, in the model through supervised neural networks made for both binary and multiclass classification tasks, utilizing, rather unpreprocessed, eigenvalue spacings and eigenvector probability densities as input features. Moreover, the output of our trained neural networks can also indicate the critical points separating ME, MBC, and MBL phases, which are consistent with the same obtained from other conventional methods. We also employ unsupervised learning techniques, particularly principal component analysis (PCA) of eigenvector probability densities, to investigate how this framework, without any training, captures the, rather unknown, many-body phases (ME, MBL, and MBC) and single-particle phases (delocalized, localized, and critical) of the interacting and noninteracting systems, respectively. Our findings reveal that PCA entropy serves as an effective indicator (order parameter) for detecting phase transitions in the single-particle systems. Moreover, this method proves applicable to many-body systems when the data undergoes a suitable preprocessing. Interestingly, when it comes to the extraction of critical (correlation length) exponents through a finite-size scaling, we find that for single-particle systems, scaling collapse of neural network outputs is obtained using components of inverse participation ratio (IPR) as input data. Remarkably, we observe identical critical exponents as obtained from the scaling collapse of the IPR directly for different single-particle phase transitions. However, the current finite-size scaling approach to the network output cannot reliably extract critical exponents for the many-body system that needs further investigation.
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.
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 demonstrate that the quantum mutual information (QMI) is a useful probe to study many-body localization (MBL). First, we focus on the detection of a metal-insulator transition for two different models, the noninteracting Aubry-André-Harper model and the spinless fermionic disordered Hubbard chain. We find that the QMI in the localized phase decays exponentially with the distance between the regions traced out, allowing us to define a correlation length, which converges to the localization length in the case of one particle. Second, we show how the QMI can be used as a dynamical indicator to distinguish an Anderson insulator phase from a MBL phase. By studying the spread of the QMI after a global quench from a random product state, we show that the QMI does not spread in the Anderson insulator phase but grows logarithmically in time in the MBL phase.