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Unsupervised machine learning account of magnetic transitions in the Hubbard model
Kelvin Ch'ng, Nick Vazquez, Ehsan Khatami
TL;DR
The paper asks whether unsupervised dimensionality reduction can locate magnetic transitions and crossovers in quantum systems, where quantum fluctuations blur the patterns seen in classical models. It applies several such methods to Monte Carlo spin configurations and finds that t-SNE indicators track antiferromagnetic structure factors in the weak-coupling regime. The authors also report a scope limit away from half filling when the quantum Monte Carlo sign problem is present.
Problem
The paper asks whether unsupervised dimensionality reduction can help locate magnetic transitions or crossovers despite quantum fluctuations blurring reduced-space patterns.
Source: B. The Fermi-Hubbard Model · p. 2, §IV · p. 5Method
The authors apply autoencoders, random trees embedding, and t-SNE to reduce auxiliary-spin configurations from finite-temperature quantum Monte Carlo simulations.
Source: §I · p. 1Results
In the weak-coupling regime, t-SNE indicators correlate perfectly with antiferromagnetic structure factors in two and three dimensions, and t-SNE patterns distinguish the 3D transition and 2D crossover.
Source: §I · p. 2, §IV · p. 9Takeaways & Limitations
Reduced-dimensional t-SNE outputs provide indicators of magnetic changes that track antiferromagnetic structure factors in the reported weak-coupling settings.
Source: §I · p. 2, §IV · p. 9The authors caution that these techniques cannot be expected to work away from half filling when the sign problem is present in quantum Monte Carlo simulations.
Source: §I · p. 2
Abstract
from arXiv · showhide
We employ several unsupervised machine learning techniques, including autoencoders, random trees embedding, and t-distributed stochastic neighboring ensemble (t-SNE), to reduce the dimensionality of, and therefore classify, raw (auxiliary) spin configurations generated, through Monte Carlo simulations of small clusters, for the Ising and Fermi-Hubbard models at finite temperatures. Results from a convolutional autoencoder for the three-dimensional Ising model can be shown to produce the magnetization and the susceptibility as a function of temperature with a high degree of accuracy. Quantum fluctuations distort this picture and prevent us from making such connections between the output of the autoencoder and physical observables for the Hubbard model. However, we are able to define an indicator based on the output of the t-SNE algorithm that shows a near perfect agreement with the antiferromagnetic structure factor of the model in two and three spatial dimensions in the weak-coupling regime. t-SNE also predicts a transition to the canted antiferromagnetic phase for the three-dimensional model when a strong magnetic field is present. We show that these techniques cannot be expected to work away from half filling when the "sign problem" in quantum Monte Carlo simulations is present.
I. INTRODUCTION
The paper applies unlabeled dimensionality-reduction methods to Monte Carlo spin configurations to study magnetic transitions and crossovers. Among the tested methods, t-SNE produces the clearest Hubbard-model patterns and indicators that track antiferromagnetic structure factors in the weak-coupling regime.
I. INTRODUCTION
- Earlier work applied supervised learning and neural networks to many-body systems, while unsupervised methods had mainly been used to classify phases of classical models.Prior examples include t-SNE clustering of 2D Ising configurations and PCA connections to magnetization and susceptibility.Source: §I · p. 1
- The study reduces unlabeled auxiliary-spin configurations from quantum Monte Carlo simulations of 2D and 3D Fermi-Hubbard models to seek features associated with magnetic observables and transitions.The analysis does not use model parameters, configuration temperatures, or transition locations as learning labels.Source: §I · p. 1
- For the 3D Ising model, a convolutional autoencoder yields indicators resembling magnetization and susceptibility, while quantum fluctuations make Hubbard-model visualizations fuzzier.The authors also report that a fully connected autoencoder combined with random trees embedding gives a roughly temperature-resolved 2D visualization.Source: §I · p. 2
- t-SNE produces distinguishable patterns across the 3D N4el transition and 2D crossover, with indicators that correlate perfectly with antiferromagnetic structure factors in the weak-coupling regime.The authors also report that t-SNE can capture a transition in a magnetic field.Source: §I · p. 2
A. 3D Ising Model
The section introduces the 3D classical Ising model and its Monte Carlo spin configurations. The model undergoes a second-order transition below 4.5J.
A. 3D Ising Model
- Below 4.5J, the 3D Ising model undergoes a second-order phase transition as temperature decreases.Source: A. 3D Ising Model · p. 2
- The Ising model uses nearest-neighbor spin products weighted by exchange strength J, with each spin taking the value ±1.J is set to 1 as the Ising model's energy unit.Source: A. 3D Ising Model · p. 2
- Monte Carlo simulations generate configurations on an 8 × 8 × 8 lattice at different temperatures, with each configuration an array of ±1 spins.Source: A. 3D Ising Model · p. 2
B. The Fermi-Hubbard Model
The study uses the Fermi-Hubbard model to examine how quantum fluctuations affect unsupervised detection of magnetic transitions and crossovers. It simulates half-filled systems in 3D and a 2D system with a finite-temperature antiferromagnetic crossover.
B. The Fermi-Hubbard Model
- The model is studied to assess how quantum fluctuations affect the ability of unsupervised methods to locate transitions or crossovers.Source: B. The Fermi-Hubbard Model · p. 2
- In the Hubbard Hamiltonian, t denotes nearest-neighbor hopping, U onsite Coulomb interaction, µ chemical potential, and h magnetic-field magnitude.By symmetry, µ = 0 gives half filling, and decreasing µ achieves hole doping.Source: B. The Fermi-Hubbard Model · p. 2
- At half filling, the 3D Hubbard model has a second-order transition to long-range N4el order below a U-dependent critical temperature.The critical temperature rises from zero at weak coupling, peaks around U = 9, and falls as 1/U at strong coupling.Source: B. The Fermi-Hubbard Model · p. 2
- DQMC simulations use a 4 × 4 × 4 lattice at U = 4, 9, and 14, saving auxiliary spin configurations across temperatures with 200 imaginary-time slices.Source: B. The Fermi-Hubbard Model · p. 2
- The 2D model has no finite-temperature transition to long-range order, but crosses over to strong antiferromagnetic correlations.For U = 4, the crossover onset is estimated around T = 0.25 and associated with a peak in uniform susceptibility.Source: B. The Fermi-Hubbard Model · pp. 2–3
A. Autoencoders
Autoencoders learn an input's reduced representation by training a network to reconstruct that input. The paper uses deeper fully connected and convolutional architectures to extract features from the data.
A. Autoencoders
- An autoencoder is an unsupervised neural network trained to reproduce its input, thereby learning features or reducing data dimensionality without prior feature distinctions.Source: A. Autoencoders · p. 3
- The coding layer provides the most reduced representation, which can support later clustering; the study uses deeper architectures and avoids PCA because it is useful only for linear correlations.A single-layer autoencoder with linear activations is similar to PCA; PCA has performed poorly for quantum relative to classical phase classification.Source: A. Autoencoders · p. 3
- Convolutional autoencoders extract features with shared filters over local data subsections, while fully connected layers connect each neuron to all neurons in adjacent layers.The study uses convolutional autoencoders to extract spatially correlated features more efficiently; decoding uses upsampling to reconstruct resolution.Source: A. Autoencoders · p. 3
B. Random Trees Embedding
Random trees embedding transforms data through an ensemble of randomized trees to extract and reduce features. Here, it is applied to low-dimensional autoencoder representations of Hubbard-model auxiliary fields.
B. Random Trees Embedding
- An ensemble of randomized trees reduces distortion from a single tree’s metric space by embedding data across overlapping random sections.Features are selected by voting on overlap density between nodes at a given depth; features below a threshold are discarded.Source: B. Random Trees Embedding · p. 3
- For the Hubbard model, random trees embedding uses 100 trees of maximum branching depth 10 on fully connected autoencoder representations.The input is a low-dimensional representation of the auxiliary-field data.Source: B. Random Trees Embedding · p. 3
C. t-SNE
The section contrasts linear PCA with nonlinear t-SNE, which aims to preserve local data structure when projecting high-dimensional data into a lower-dimensional space. The chosen perplexity is about 27, while t-SNE scales quadratically with the number of data points.
C. t-SNE
- PCA maximizes projected variance through a linear projection, so it does not preserve local structures with nonlinear correlations.The text notes that linearity can make simple Ising observables such as magnetization apparent, but cannot be assumed for auxiliary-field observables.Source: C. t-SNE · p. 4
- t-SNE uses a Student-t distribution in the low-dimensional space and a cost function based on similarity distributions to preserve local structure.Its predecessor SNE used Gaussian distributions and adjusted their widths to maintain a user-specified effective neighbor count, or perplexity.Source: C. t-SNE · p. 4
- A perplexity of about 27 produced the most physically interesting features across the tested interaction strengths, within the suggested range of 5 to 50.The method scales as O(n^2) in the number of data points.Source: C. t-SNE · p. 4
D. k-means
k-means identifies clusters by iteratively assigning points to the nearest of k user-specified centroids and updating those centroids. Here it quantifies temperature-dependent spread in two-dimensional autoencoder or t-SNE outputs.
D. k-means
- k-means iteratively assigns each point to its nearest centroid and recalculates centroids until they stabilize.It is most effective when the data form well-separated clusters.Source: D. k-means · p. 4
- The study applies k-means to two-dimensional autoencoder or t-SNE outputs to quantify data spread at each temperature.It requests three centroids at each temperature and defines D as their mean distance from their common center.Source: D. k-means · p. 4
- D is not a unique indicator; other measures may better capture how output features evolve with temperature.Source: D. k-means · p. 4
IV. RESULTS
Across the tested Hubbard-model settings, t-SNE gives the clearest temperature-dependent patterns and, in weak coupling, an indicator that tracks antiferromagnetic structure factors in two and three dimensions. The Ising benchmark connects reduced representations to magnetization and susceptibility, while the Hubbard autoencoder and random-tree results are less directly interpretable as observables.
IV. RESULTS
- For the 3D Ising benchmark, the one-neuron autoencoder output bifurcates below Tc and resembles magnetization, while k-means spread D resembles susceptibility.The fitted critical temperature is B = 4.55J, compared with 4.54J estimated for a system of the same size; the fitted exponent β = 0.34 is close to 0.33.Source: §IV · p. 5
- For Hubbard autoencoders, low-temperature clustering is not distinct as in the Ising case, and the k-means indicator does not agree well with the AFM structure factor at low temperatures.The indicator is noisy for U = 14; the authors report that this approach is most efficient at locating critical behavior in weak coupling.Source: §IV · pp. 5–6, Figure 2 · p. 6
- Combining a convolutional autoencoder with random trees embedding produces low-temperature clusters for U = 4 and 9, while fully connected autoencoder inputs resolve temperature more clearly but show no phase-specific feature.The convolutional autoencoder’s convolutions appear important for low-temperature clustering, but not for temperature resolution.Source: §IV · pp. 6–7, Figure 3 · p. 6
- In 3D t-SNE outputs, U = 4 and 9 show temperature-separated patterns, while U = 14 forms worm-like structures attributed to Mott physics and less-ergodic simulations.At U = 4, low-temperature clusters are visible, though some points remain scattered between their centers.Source: §IV · p. 7
- In the weak-coupling 2D Hubbard model at half filling, the t-SNE indicator D closely tracks the AFM structure factor at U = 4 and rises sharply below T = 0.2.The study reports no specific features pointing to a crossover.Source: §IV · pp. 7–8
- With a magnetic field h = 1.0, t-SNE identifies a low-temperature region that shrinks below T ∼ 0.3, signaling a phase change; the autoencoder indicator is flat and misses changes in the xy plane.The field mostly aligns spins along z while remnant AFM correlations lie in the xy plane.Source: §IV · p. 8
- Away from half filling, ignoring configuration signs yields no physically meaningful indicator for the Hubbard model when the DQMC sign problem is present.The measured structure factor depends on the difference between positive- and negative-sign contributions divided by the average sign; at µ < −1.0, sign-separated scatter plots look essentially alike.Source: §IV · pp. 8–9
- The paper concludes that unsupervised reductions can extract features resembling magnetic observables at and away from half filling, but those measures cannot serve that purpose away from half filling in the presence of the sign problem.Source: §IV · pp. 9–10