PhaseLearner

Machine learning methods for detecting phase transitions in the optical SSH (Su-Schrieffer-Heeger) model. Four unsupervised/semi-supervised approaches are benchmarked against finite-size scaling (FSS) reference values for the valence bond solid (VBS) critical inverse temperature $\beta_c$.

Physical Context

All models are use an $L \times L$ lattice, with $L =12$, filled with bond-bond correlation data in momentum space. The key parameters are:

• $\alpha$ — coupling constant (0.240, 0.245, 0.250, 0.255)
• $\beta$ — inverse temperature; the goal is to identify the critical value β_c where the VBS phase transition occurs
• $\omega= 0.5$ — phonon frequency

All parameters are in units of $t$, the hopping amplitude of electrons appearing in the kinetic term of the Hamiltonian.

FSS reference values:

$\alpha$	$\beta_c$
0.240	17.5
0.245	14.3
0.250	12.5
0.255	12.3

Methods

1. PCA (pca_method.py)

Principal Component Analysis on the $L^2$-dimensional bond-bond correlator configurations. $\beta_c$ is estimated from:

• PCA susceptibility Peak of $Var(P_1)$ and $Var(P_2)$ as a function of $\beta$ (analogous to magnetic susceptibility)
• KMeans clustering $k=2$ clustering in $(P_1, P_2)$ space; sigmoid fit to the KVBS fraction $f_{KVBS}(\beta)$

2. UMAP (umap_method.py)

Non-linear dimensionality reduction to a $2$-dimensional embedding, followed by:

• KMeans clustering and sigmoid fit to $f_{KVBS}(\beta)$
• GMM (Gaussian Mixture Model) clustering as a secondary estimate

3. Assaad / Greitemann–Iqbal method (assaad_method.py)

CNN-based approach: trains a classifier to distinguish configurations on either side of a guessed $\beta_c$, then sweeps over candidate values. The true $\beta_c$ maximizes test accuracy. Needs significantly more samples than the rest of the methods!

4. Autoencoder (autoencoder.py)

An autoencoder with a $2$-dimensional bottleneck (latent_dim=2) trained on all configurations. $\beta_c$ is estimated from:

• Reconstruction error The reconstruction error is highest at $\beta_c$ since ambiguity is notable accross many samples
• Latent KMeans $k=2$ clustering in latent space $(z_0, z_1)$ with sigmoid fit

Installation

pip install -e .

For UMAP support:

pip install -e ".[umap]"

Requirements: Python ≥ 3.8, numpy, scipy, scikit-learn, torch, matplotlib. Optional: umap-learn.

Data

Place input CSV files in a NEW/ directory at the project root. Each file should be named:

BondBond012345_momentum_L12_w0.5_m0.0_a{alpha:.4f}-1.csv

Each row contains $144$ momentum-space features (flattened $12\times12$ lattice) followed by the inverse temperature $\beta$.

Usage

Run all methods

python -m ssh_ml_phase

Run individual methods

python -m ssh_ml_phase.pca_method
python -m ssh_ml_phase.umap_method
python -m ssh_ml_phase.assaad_method
python -m ssh_ml_phase.autoencoder

Single alpha (fast iteration)

python -m ssh_ml_phase.pca_method --alphas 0.240

Skip slow methods

python -m ssh_ml_phase --skip-assaad --skip-ae

Results are saved as .npz files in the results/ directory.

Visualization

Open SSH_ML_phase_detection.ipynb after running the methods. The notebook loads the pre-computed .npz result files and generates comparison plots saved to ssh_ml_phase/figures/:

Figure	Content
PCA_susceptibility_argmax.pdf	$Var(P_1)$, $Var(P_2)$, and their derivatives vs $\beta$
PCA_kmeans.pdf	$2D$ PCA embedding and KMeans $f_{KVBS}$ curves
UMAP_beta_c.pdf	UMAP embedding and KMeans/GMM $f_{KVBS}$ curves
Part4_autoencoder.pdf	Mean reconstruction error, derivative, and latent order parameter
Part4_latent_scatter.pdf	Autoencoder latent space scatter coloured by $\beta$ and cluster
Final_summary_all_methods.pdf	All methods' $\beta_c$ estimates vs FSS reference

Project Structure

PhaseLearner/
├── ssh_ml_phase/
│   ├── config.py          # FSS references, paths, physical constants
│   ├── data.py            # CSV loading utilities
│   ├── pca_method.py      # PCA susceptibility and KMeans
│   ├── umap_method.py     # UMAP + KMeans/GMM
│   ├── assaad_method.py   # Greitemann–Iqbal CNN method
│   ├── autoencoder.py     # Autoencoder anomaly detection
│   ├── run_all.py         # Orchestrates all methods
│   └── figures/           # Output plots
├── results/               # Saved .npz result files (generated)
├── NEW/                   # Input CSV data (not tracked)
├── SSH_ML_phase_detection.ipynb
└── setup.py