Source-linked AI summary
Probing many-body localization with neural networks
Frank Schindler, Nicolas Regnault, Titus Neupert
TL;DR
Distinguishing thermalizing ETH states from many-body localized states remains a central question, and established criteria rely on physical understanding of the phases. This paper trains a neural network on entanglement spectra of the random-field Heisenberg chain and applies it to states across the transition. Its phase diagram agrees with conventional methods, while multiple real-space cuts support spatially resolved classification of eigenstates.
Problem
Conventional ETH–MBL criteria rely on physical understanding of the phases, while their characteristics may not be fully understood.
Source: §I · p. 2Method
The authors train a neural network on entanglement spectra and classify eigenstates of the random-field Heisenberg chain using spectra from multiple real-space cuts.
Source: §I · p. 2Results
The resulting phase diagram agrees well with conventional methods and locates the transition as a function of energy density and disorder strength.
Source: §I · p. 2Takeaways & Limitations
The approach can characterize individual eigenstates across the transition with spatial resolution without requiring a priori knowledge of phase-discriminating criteria.
Source: §I · p. 2
Abstract
from arXiv · showhide
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.
I. INTRODUCTION
This work uses entanglement spectra to classify ETH and MBL states in the random-field Heisenberg chain, without requiring a priori criteria for distinguishing the phases. It finds a phase diagram consistent with conventional methods and applies the classifier to individual eigenstates and real-space cuts.
I. INTRODUCTION
- Unlike approaches trained on prototype configurations, this method uses entanglement spectra to classify phases and transitions.Entanglement spectra had also been used for neural-network detection of phase transitions.Source: §I · p. 1
- The study applies this approach to the random-field Heisenberg chain, which is thermalizing at weak disorder and localized at strong disorder.The chain serves as a model for many-body localization.Source: §I · p. 1
- The method requires only that distinguishing information be present in the entanglement spectrum, rather than prior knowledge of physical criteria for ETH or MBL.The authors identify this as useful when phase characteristics are not fully understood.Source: §I · p. 2
- The network’s phase diagram locates the ETH–MBL transition across energy density and disorder strength, in good agreement with conventional methods.The diagram averages classifications over disorder realizations.Source: §I · p. 2
- Using spectra from multiple real-space cuts lets the network trace individual eigenstates and map their classification across the transition.The study considers all eigenstates, including those at finite energy density.Source: §I · p. 2
II. NEURAL NETWORKS FOR BINARY CLASSIFICATION
The section introduces neural networks as trainable maps for binary classification, then describes training, validation, and the network’s layer operations. The output is a continuous pair of class confidences that is trained against labeled examples.
II. NEURAL NETWORKS FOR BINARY CLASSIFICATION
- For binary classification, inputs are mapped to one of two one-hot target vectors, each with a single nonzero entry.The network’s output vector is continuous even though the target set is discrete.Source: §II · p. 2
- The network learns a continuous trial map from labeled examples, then evaluates generalization on held-out data while cross-validation averages outputs across training runs.The data is randomly split into equally sized training and testing sets.Source: §II · pp. 2–3
- Each layer applies an affine map and nonlinear activation to transform its input vector into the next layer’s neuron outputs.The first layer maps an input of dimension n1 to a vector of dimension n2.Source: §II · p. 2
- The chosen architecture has one hidden layer, with ReLU activation and a Softmax output whose two class confidences sum to one.The output neurons represent the two classification options.Source: §II · pp. 2–3
- Training adjusts network weights and biases to minimize a discrepancy functional, using cross-entropy as the common choice for Softmax outputs.Stochastic gradient descent uses small randomly selected subsets of training data for its iterations.Source: §II · p. 3
III. MANY-BODY LOCALIZATION IN THE HEISENBERG CHAIN AND ENTANGLEMENT SPECTRUM
The section sets up the random-field Heisenberg chain and defines the entanglement spectrum used to distinguish ETH and MBL behavior. It reviews established entanglement-based criteria and motivates a neural-network classifier that learns discriminating features from examples.
III. MANY-BODY LOCALIZATION IN THE HEISENBERG CHAIN AND ENTANGLEMENT SPECTRUM
- The random-field Heisenberg chain is thermalizing at weak disorder and many-body localized at strong disorder, with different energy-level statistics in the two regimes.The ETH regime exhibits GOE level repulsion, whereas the MBL regime has Poisson statistics.Source: §III · p. 3
- Near the transition, whether a system is classified as ETH or MBL can depend on both its disorder realization and the eigenstate’s energy density.Averaging over disorder removes realization-specific dependencies, but the transition may still vary with energy density.Source: §III · p. 3
- The entanglement spectrum is the spectrum of the entanglement Hamiltonian He, defined through the reduced density matrix for a real-space partition.The system is divided into regions A and B, and the degrees of freedom in B are traced out.Source: §III · p. 3
- Existing entanglement criteria include the Schmidt gap, entropy scaling, entropy fluctuations across energy windows, and entanglement-spectrum level statistics.The cited criteria distinguish regimes through near-purity, area- or volume-law scaling, enhanced fluctuations near the transition, or different level distributions.Source: §III · pp. 3–4
- The neural-network approach does not require prior knowledge of these criteria; it is expected to learn them from labeled examples.The authors compare its performance with entanglement-based measures and energy-level statistics.Source: §III · p. 4
IV. TRAINING DATA AND NETWORK ARCHITECTURE
A single-hidden-layer network classifies exact-diagonalization entanglement spectra from the disordered Heisenberg chain. Training uses endpoint regimes, regularization, and a confidence term for unlabeled intermediate-disorder spectra.
IV. TRAINING DATA AND NETWORK ARCHITECTURE
- Training uses entanglement spectra of exact-diagonalized eigenstates, focusing on the Sz tot = 0 sector and the largest entanglement-Hamiltonian block.For each cut, the spectrum has length n1 = 2^NA before block diagonalization.Source: §IV · p. 4
- Training excludes the lowest and highest 10% of energy states and uses only the lower half of each entanglement spectrum.The upper half is generally below 10^-16 in both regimes, where finite machine precision makes the eigenvalues uninformative.Source: §IV · p. 4
- Dropout randomly trains with half the hidden units, while weight decay applies only to hidden-layer weights to preserve a reparametrization symmetry.Dropout is described as limiting adaptation to nonuniversal training-data features; weight decay suppresses weights no longer contributing to error minimization.Source: §IV · pp. 4–5
- Testing accuracy is η = 1, including when training and testing spectra come from different disorder realizations.The testing samples do not overlap with the training data.Source: §IV · p. 5
- Confidence optimization penalizes uncertain predictions on intermediate-disorder spectra, and both its strength δ and weight decay µ are set to 1.The transition-region spectra are unlabeled, and the authors report that order-one choices of both hyperparameters do not significantly affect results.Source: §IV · p. 5
- The network trains on labeled spectra at h̄ = 0.25 and h̄ = 12.0, then classifies unlabeled spectra at intermediate disorder.The transition-region set makes no assumption about whether those states are ETH or MBL.Source: §IV · p. 5
A. Disorder-averaged phase diagram
The network reproduces the disorder-averaged phase diagram and yields a critical disorder estimate under an explicit confidence and state-fraction criterion.
A. Disorder-averaged phase diagram
- h̄c = 2.8 ± 0.5 when 90% confidence defines an MBL spectrum and half the spectra at a given h̄ must meet that threshold.The estimate agrees with prior studies.Source: A. Disorder-averaged phase diagram · p. 6
B. Single disorder realization
For a single disorder realization, the network’s transition estimate agrees with established indicators and is sharper, while its individual-state classifications are clearer than the Schmidt-gap criterion. Confidence optimization is important for agreement with energy-level statistics.
B. Single disorder realization
- The single-sample analysis averages over multiple training runs because randomized initialization can slightly alter individual-state classifications.The authors characterize the average deviation as negligible.Source: B. Single disorder realization · p. 6
- The neural network’s transition shape and location agree very well with the von Neumann entanglement-entropy standard deviation and absolute value of the h̄ derivative of the Schmidt gap, while yielding a much sharper transition.Source: B. Single disorder realization · p. 6
- Individual eigenstates are classified more clearly by the network than by the Schmidt-gap criterion.The fine-structure comparison uses representative eigenstates from one disorder sample.Source: B. Single disorder realization · p. 6
- Confidence optimization makes the network’s transition characterization align with established energy-level statistics; omitting it produces a large deviation.The added cost term favors confident classification of transition-region states.Source: Figure 5 · p. 6, B. Single disorder realization · p. 7
- The same or better classification can be achieved by refining and combining conventional methods, though the network requires little prior knowledge.The authors describe the network as a basic single-hidden-layer model with standard regularization; its nontrivial addition is cost optimization with nonzero δ.Source: B. Single disorder realization · p. 7
- The transition location varies substantially across disorder realizations, with mean absolute field strength correlated with its inverse.The passage notes that other disorder properties may also correlate with the transition location.Source: B. Single disorder realization · p. 7
C. Spatial structure of individual eigenstates
The network classifies eigenstates across seven entanglement cuts to resolve their local character as disorder varies. The results reveal an asymmetry between approaching the transition from the ETH and MBL sides, with open boundary conditions required for cuts to provide purely local information.
C. Spatial structure of individual eigenstates
- For a fixed disorder realization, the study classifies eigenstate entanglement spectra across seven consecutive middle-chain cuts; each cut requires a separately trained network.The analysis includes only confidently classified spectra and does not use cross-validation.Source: C. Spatial structure of individual… · p. 7
- Figure 6(a) shows classifications of 200 middle-spectrum eigenstates across cuts with N_A = 6 to 12, while panel (b) surveys 10,000 such eigenstates and marks states with mixed ETH/MBL classifications across cuts.Blue and red indicate confident ETH and MBL classifications; white indicates the remaining states.Source: Figure 6 · p. 7
- A substantial fraction of states classified as MBL at some cut deep in the ETH regime remain MBL across all seven cuts, while only a tiny fraction of states classified as ETH deep in the MBL regime remain ETH across all cuts.The contrast points to asymmetric evolution of local eigenstate structure as the transition is approached from either side.Source: C. Spatial structure of individual… · p. 7
- As the transition is approached from the MBL side, ETH-like regions emerge in rare locations with small random-field variations, grow, and can eventually act as a bath for the whole system.The resulting states can become entirely ETH-like.Source: C. Spatial structure of individual… · p. 7
- The local interpretation of an entanglement cut relies on open boundary conditions; with periodic boundaries, two distant cuts would mix information from both locations.Source: C. Spatial structure of individual… · p. 7
D. What the network has learned: dreaming
The authors use dreaming to probe which entanglement-spectrum features the trained classifier has learned. Dreamed spectra recover the relative characteristic shapes of ETH and MBL spectra, including the MBL power law, without reproducing their absolute magnitudes.
D. What the network has learned: dreaming
- Dreaming starts from transition-region spectra classified with 40%–60% confidence and optimizes them toward ETH or MBL outputs while enforcing nonnegative reduced-density-matrix eigenvalues and ordered absolute logarithms.The algorithm uses cross-entropy and does not require the eigenvalues to sum to one.Source: D. What the network has learned:… · p. 8
- Dreamed spectra reproduce the relative characteristic shapes of typical ETH and MBL spectra, including the MBL phase’s power-law entanglement spectrum, but not their exact absolute magnitudes.The network was trained to distinguish phases, not to characterize either phase individually.Source: D. What the network has learned:… · p. 8
- The authors average over 40 network trainings and use 200,000 gradient-descent steps per dreaming run; Figure 7 compares the resulting spectra with typical ETH, MBL, and transition spectra.The figure’s horizontal axis indexes entanglement-spectrum eigenvalues.Source: D. What the network has learned:… · p. 8, Figure 7 · p. 8
VI. SUMMARY
The paper uses a single-hidden-layer neural network trained on entanglement spectra from the ETH and MBL regimes to classify states between them. It reports agreement with previous phase diagrams and especially clean classification of individual eigenstates from one disorder realization, while noting that the method is uncontrolled for quantitative critical-disorder estimates.
VI. SUMMARY
- The network’s phase diagram agrees with previous studies, and its cleanest characterization is for individual eigenstates in a single disorder realization rather than quantitative estimation of critical disorder.Dreaming indicates that it learns the MBL entanglement spectrum’s power-law structure.Source: §VI · p. 8
- A single-hidden-layer network trained on ETH- and MBL-regime spectra is applied to states between those regimes, although it was not trained on transition-region spectra.Source: §VI · p. 8
- The authors describe the network structure and operation, including confidence optimization and dreaming, as broadly applicable to classifying phases from entanglement spectra and possibly other correlation functions.Source: §VI · p. 8
- Without proper regularization, network performance becomes pathological; omitting confidence optimization shifts the detected transition, and the authors report that the full cost function better matches level-statistics results.Source: Figure 8 · p. 9
Appendix A: Training with other kinds of input data
The appendix tests alternative inputs and reports that the phase diagram remains in quantitative agreement with the main-text result when both regularization and confidence optimization are retained. It also describes the network settings and the study’s procedures for comparing disorder realizations and training runs.
Appendix A: Training with other kinds of input data
- For reduced-density-matrix spectra, the single-hidden-layer network reaches η = 1 and uses the first few inputs, rediscovering the Schmidt-gap criterion; for eigenvalue differences, the single-layer network does not converge to η = 1 in a reasonable number of steps, but a second hidden layer does.Source: Appendix A · p. 9
- The appendix compares 50 training runs per disorder realization and examines transition-region shapes across N = 12, 14, 16, and 18, using different disorder realizations and averaging eigenstates to control visual resolution.The N = 16 and N = 18 plots use the same number of averaged eigenstates; transition locations should not be compared across samples.Source: Figure 9 · p. 9, Figure 10 · p. 10
- With either reduced-density-matrix spectra or differences of entanglement-spectrum eigenvalues as input, the phase diagram agrees quantitatively with the main-text result when regularization and confidence optimization are both included.Dropping either term makes the phase diagram input-dependent, consistent with inequivalent minima of the naive cost function.Source: Appendix A · pp. 9–10
Appendix B: Hyperparameters and network details
- For inputs of size at most 126, the network uses 1,024 hidden units; training uses gradient-descent step size 10^-4, μ = 1, δ = 1, batches of 100, and 3,000–4,000 steps.The parameters were chosen empirically, and the authors note that hidden-unit count is somewhat arbitrary if sufficiently large relative to input size.Source: Appendix B · p. 10
- Ignoring either confidence optimization or weight decay yields classification inconsistent with other methods.Source: Appendix B · p. 10
Appendix C: Empirical observations on the network performance
Repeated training runs produce mostly consistent state classifications, though confidence assignments vary more than whether a state is classified as ETH or MBL. The appendix measures this variation across 50 runs on one N = 18 disorder realization.
Appendix C: Empirical observations on the network performance
- The study trained the single-hidden-layer network 50 times for 3000 steps on entanglement spectra from one N = 18 disorder realization.The observed inconsistency stemmed mainly from random weight and bias initialization, rather than random training and testing sets.Source: Appendix C · p. 10
- A state counted as MBL only above 90% MBL confidence and as ETH below 10%, with Softmax outputs summing to 100%.States between the thresholds were not confidently classified as either phase.Source: Appendix C · p. 10
- 0.008 of states changed between ETH and MBL classifications across run pairs, while 0.07 were confidently classified in one run but not the other.These are average fractions across all possible pairings of 50 trained phase diagrams.Source: Appendix C · p. 11