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.
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.
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.
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 apply support vector machine (SVM) to study the phase transition between many-body localized and thermal phases in a disordered quantum Ising chain in a transverse external field. The many-body eigenstate energy E is bounded by a bandwidth $W={E}_{\text{max}}\ensuremath{-}{E}_{\text{min}}$. The transition takes place on a phase diagram spanned by the energy density $\ensuremath{\epsilon}=2(E\ensuremath{-}{E}_{\text{min}})/W$ and the disorder strength $\ensuremath{\delta}J$ of the spin interaction uniformly distributed within $[\ensuremath{-}\ensuremath{\delta}J,\ensuremath{\delta}J]$, formally parallel to the mobility edge in Anderson localization. In our paper, we use the labeled probability density of eigenstate wave functions belonging to the deeply localized and thermal regimes at two different energy densities ($\ensuremath{\epsilon}$'s) as the training set, i.e., providing labeled data at four corners of the phase diagram. Then we employ the trained SVM to predict the whole phase diagram. The obtained phase boundary qualitatively agrees with previous work using entanglement entropy to characterize these two phases. We further analyze the decision function of the SVM to interpret its physical meaning and find that it is analogous to the inverse participation ratio in configuration space. Our findings demonstrate the ability of the SVM to capture potential quantities that may characterize the many-body localization phase transition.
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 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 investigate a one-dimensional tight-binding model in which onsite potentials [Formula: see text] exhibit power-law spatial correlations (with exponent α) and the hopping amplitudes decay as [Formula: see text]. This two-parameter family interpolates continuously between short-range Anderson-like disorder, correlated disorder with conventional hopping, and long-range hopping models with nontrivial delocalization tendencies. Using large-scale exact diagonalization, we construct a comprehensive phase map in the [Formula: see text] plane by combining spectral statistics, density-of-states analysis, and energy-resolved localization indicators such as the participation ratio, single-particle entanglement entropy, level-spacing ratio r, and the ratio of the geometric to arithmetic density of states. From these observables we define phase-indicator functions that compactly quantify localization behavior across the spectrum. Our analysis reveals robust mobility edges and multiple regimes of spectral coexistence between localized, extended, resonant, and critical states. Finite-size scaling, implemented via an explicit smoothness-based cost function, enables extraction of critical exponents and delineation of transition lines across the [Formula: see text] parameter space. To validate and complement these physics-based diagnostics, we employ a supervised autoencoder that learns high-level representations of eigenstate structure directly from raw features and reliably reproduces the phase classification defined by the indicator functions. Together, these approaches provide a coherent and internally consistent picture of the spectral transitions driven by correlated disorder and long-range hopping, establishing a unified framework for characterizing mobility edges in long-range one-dimensional systems.