Source-linked AI summary
Learning phase transitions by confusion
Evert P. L. van Nieuwenburg, Ye-Hua Liu, Sebastian D. Huber
TL;DR
Identifying phase transitions is difficult when quantum wavefunctions are exponentially large and phase labels or order parameters are unavailable. The paper compresses quantum states with entanglement spectra and trains neural networks on deliberately mislabeled data, using performance to locate transitions across three models. It reports successful detection of the Kitaev-chain, classical-Ising, and disordered-spin-chain transitions without relying on phase-specific knowledge.
Problem
Phase-transition classification can involve exponentially large quantum wavefunctions, while the labels to classify phases are often unknown beforehand.
Method
The confusion scheme purposefully assigns labels according to proposed critical points and evaluates neural-network performance, using entanglement spectra as quantum-state inputs.
Results
The method identifies transitions in the Kitaev chain, classical Ising model, and disordered quantum spin chain, including a W-shaped performance function for the tested cases.
Takeaways & Limitations
The approach can identify phase transitions without prior knowledge of the phases and may help analyze cases where phase identity or intervening phases are unclear.
Takeaways & Limitations
The presented scheme assumes that two phases are being distinguished; multiple transitions would modify the characteristic W-shape.
Abstract
from arXiv · showhide
Classifying phases of matter is a central problem in physics. For quantum mechanical systems, this task can be daunting owing to the exponentially large Hilbert space. Thanks to the available computing power and access to ever larger data sets, classification problems are now routinely solved using machine learning techniques. Here, we propose to use a neural network based approach to find phase transitions depending on the performance of the neural network after training it with deliberately incorrectly labelled data. We demonstrate the success of this method on the topological phase transition in the Kitaev chain, the thermal phase transition in the classical Ising model, and the many-body-localization transition in a disordered quantum spin chain. Our method does not depend on order parameters, knowledge of the topological content of the phases, or any other specifics of the transition at hand. It therefore paves the way to a generic tool to identify unexplored phase transitions.
I. INTRODUCTION
The paper combines supervised and unsupervised learning to detect phase transitions when phase labels are unknown, using deliberately incorrect labels to evaluate neural-network performance. For quantum systems, it uses the entanglement spectrum to compress exponentially large wavefunctions.
- I. INTRODUCTION: The method is motivated by machine learning’s growing use for analyzing increasingly large datasets with improved computing power.The paper situates its approach within broader supervised and unsupervised machine-learning methods.
- I. INTRODUCTION: The proposed confusion scheme purposefully mislabels data and identifies phase labels by evaluating neural-network performance.It combines supervised and unsupervised techniques because the labels to be discovered are not known beforehand.
- I. INTRODUCTION: The entanglement spectrum compresses quantum wavefunction data while retaining important information about the underlying state.The approach uses the entanglement spectrum because quantum wavefunctions contain exponentially many coefficients as system size increases.
- I. INTRODUCTION: The Kitaev chain provides a demonstration setting with analytically available results and a clear entanglement-spectrum distinction between phases.The study also introduces blanking and the characteristic performance shape before applying the methodology to more complex models.
II. RESULTS
The Kitaev chain undergoes a transition between topologically trivial and nontrivial ground states, and its entanglement spectrum provides a compact representation for machine-learning analysis. The spectrum is visibly distinguishable across the two phases.
- II. RESULTS: The Kitaev chain transitions from a topologically trivial state for |µ| > 2t to a nontrivial state for |µ| < 2t at µ = ±2t.Here, t controls both hopping and pairing of spinless fermions, while µ is the chemical potential.
- II. RESULTS: The entanglement spectrum is constructed from a spatial bipartition into left and right halves after tracing out one subset.It is defined from the eigenvalues λ_i of the reduced density matrix as the set of numbers −ln λ_i.
- II. RESULTS: The two Kitaev-chain phases are clearly distinguishable in the entanglement spectrum, particularly through degeneracy in the nontrivial phase.The paper uses this model as a demonstration before applying the method to more complex systems.
- II. RESULTS: The machine-learning inputs are the largest 10 eigenvalues for L = 20 with equal partitions LA = LB = 10 over −4t ≤ µ ≤ 0.
A. Unsupervised learning
Principal component analysis extracts orthogonal directions of maximal variance from the entanglement spectra and separates the Kitaev-chain data into three groups around the transition.
- A. Unsupervised learning: PCA transforms the 10-dimensional entanglement spectra into orthogonal axes that capture most of the data variance.The transformation is Y = XW, where X contains the spectra as rows.
- A. Unsupervised learning: The principal components are determined from the eigenvalue equation X^T Xω_ℓ = λ_ℓω_ℓ.The eigenvalues are non-negative and normalized so their sum is one.
- A. Unsupervised learning: The PCA projection clusters the spectra into µ < −2t, µ = −2t, and µ > −2t.
B. Supervised learning
A feedforward neural network learns phase labels from entanglement spectra and generalizes to held-out data. When training data near the transition is blanked, it can still interpolate the transition point within tested widths.
- B. Supervised learning: The neural network is included because PCA is linear and may not extract nonlinear relationships in more complex models.The proposed confusion scheme combines PCA’s unsupervised setting with neural networks’ nonlinear fitting ability.
- B. Supervised learning: The network uses 80 hidden sigmoid neurons and 2 output neurons to predict trivial and nontrivial phase probabilities.It is trained with stochastic gradient descent, L2 regularization, and a cross-entropy cost function.
- B. Supervised learning: The network correctly predicts unseen data after training on only a subset of the samples.This demonstrates generalization beyond the examples encountered during training.
C. Confusion scheme
The confusion scheme deliberately varies a proposed phase boundary, trains on the resulting labels, and identifies the transition from a universal W-shaped performance curve. It recovers the Ising transition and distinguishes structured data from unstructured random data.
- Confusion scheme: The scheme labels samples on either side of a proposed critical point c′, evaluates the trained network on all data, and uses P(c′) to locate the transition.The performance function is predicted to have a W-shape whose middle peak corresponds to the correct critical point.
- Confusion scheme: The W-shape arises because incorrectly splitting a shared feature forces the network toward the majority label, reducing performance between the true and proposed critical points.At the endpoints and at the correct labeling, the network can classify both structures consistently.
- Classical Ising model: For the two-dimensional Ising model, the W-shape predicts the correct transition temperature, with sharper underlying features producing better performance at L = 20.For L = 10, the middle peak deviates from Tc because of finite-size effects; for L = 20, it is located at the exact thermodynamic transition temperature kBTc ≈2.27J.
- Validation: The middle peak disappears and becomes a V-shape when the network is trained on unstructured random data with no valid labeling.This test indicates that the W-shape depends on extracting structure from the input data.
- Training conditions: Successful training depends essentially on the learning rate and L2 regularization, although overfitting is not necessarily harmful for identifying majority labels.The authors report that even negative L2 regularization can produce equally good performance under some training conditions.
- Many-body localization: For the many-body-localization data, the performance curve is robust near h′c ≈3J across training conditions, supporting that labeling.The figure reports batch size Nb = 100, learning rate α = 10−8, and regularization l2 = 0.01 for the displayed result.
D. Random-field Heisenberg chain
The random-field Heisenberg model exhibits a disorder-driven transition between thermalizing and many-body-localized behavior. PCA does not clearly cluster its entanglement spectra, whereas the confusion scheme detects the transition near hc ≈3J.
- Model and transition: The model transitions between thermalizing and many-body-localized behavior as the disorder strength hmax varies.Energy-level statistics and entanglement-spectrum statistics change across the transition.
- Model and transition: The study samples disorder strengths from hmax = J to hmax = 5J without relying on prior knowledge of the transition location.At each disorder strength, 10^5 realizations are generated for system size L = 12.
- Unsupervised analysis: PCA of the entanglement spectra does not reveal clear clustering in the first two principal components.The remaining components are of order 10−4 or less, but the projected data still lacks visibly separated groups.
- Confusion-scheme result: A shallow feedforward network trained on 64-level entanglement spectra produces the characteristic W-shape and detects the transition at hc ≈3J.The network uses 64 input neurons, 100 hidden neurons, and 2 output neurons.
- Confusion-scheme result: The network’s performance is also evaluated at the proposed transition point to test whether the corresponding labeling remains robust under parameter changes.This robustness check is presented as additional confirmation of the h′c ≈3J labeling.
III. DISCUSSION
The method identifies phase transitions by confusing a neural network with deliberately incorrect labels, without prior knowledge of the phases. Its two-phase assumption remains a target for future extension to multiple phases and adaptive labeling.
- The method identifies transitions between two phases by deliberately mislabeling data and evaluating overall neural-network performance.
- Using entanglement spectra compresses exponentially large quantum wavefunctions into inputs for phase-transition detection.
- The approach may identify hidden patterns distinguishing phases even when phase identity or intervening phases are unclear.
- The current framework assumes two phases; multiple transitions would modify the characteristic W-shape, with peak count indicating the number of labels.
- Future work could use adaptive labeling so the algorithm determines the correct labels autonomously.