Source-linked AI summary
Identifying topological order through unsupervised machine learning
Joaquin F. Rodriguez-Nieva, Mathias S. Scheurer
TL;DR
Identifying topological invariants from bare data is challenging because conventional distance-based approaches are unsuitable for topological classification. The paper uses unsupervised diffusion maps, which separate topological sectors, reveal their natural ordering, and identify a transition near Tc/J = 0.9 ± 0.1.
Problem
Identifying topological invariants from bare data is challenging, and Euclidean distances are unsuitable for topological classification.
Method
The paper uses diffusion maps to learn topological information without supervision from a high-dimensional feature space extracted from the data.
Results
Diffusion maps separate topological sectors, reveal a monotonic natural ordering, and locate the critical temperature at Tc/J = 0.9 ± 0.1.
Takeaways & Limitations
The approach can learn topological band indices without supervision and identify when topological sectors cease to be well defined.
Takeaways & Limitations
The comparison approach has serious drawbacks because its success depends on choosing correctly, while Euclidean distances are unsuitable for topological classification.
Abstract
from arXiv · showhide
The Landau description of phase transitions relies on the identification of a local order parameter that indicates the onset of a symmetry-breaking phase. In contrast, topological phase transitions evade this paradigm and, as a result, are harder to identify. Recently, machine learning techniques have been shown to be capable of characterizing topological order in the presence of human supervision. Here, we propose an unsupervised approach based on diffusion maps that learns topological phase transitions from raw data without the need of manual feature engineering. Using bare spin configurations as input, the approach is shown to be capable of classifying samples of the two-dimensional XY model by winding number and capture the Berezinskii-Kosterlitz-Thouless transition. We also demonstrate the success of the approach on the Ising gauge theory, another paradigmatic model with topological order. In addition, a connection between the output of diffusion maps and the eigenstates of a quantum-well Hamiltonian is derived. Topological classification via diffusion maps can therefore enable fully unsupervised studies of exotic phases of matter.
Extended data set
Diffusion maps classify highly distorted spin configurations by winding number, including seven sectors, while also recovering their natural ordering. The method learns the topological band index without supervision from the same data representation.
- Extended data set: Diffusion maps learn the topological band index without supervision when the spin-site index is viewed as discretized momentum.The same construction connects one-dimensional spin configurations to a two-band topological insulator.
- Extended data set: Seven-fold degeneracy of the largest diffusion-map eigenvalue matches the seven winding-number sectors.The samples use winding numbers {−3, −2, −1, 0, 1, 2, 3}.
- Extended data set: The first diffusion-map component separates all samples according to winding number.The clustering remains effective despite strong deformations and local unwinding of spin configurations.
- Extended data set: A monotonic relation between winding number and ψ1 reveals a natural ordering of the sectors, not merely their clustering.This ordering follows from diffusion-map distances approximating diffusion distances in the low-dimensional feature space.
- Extended data set: 100% fidelity is achieved when k-means is applied to the six-dimensional reduced diffusion-map feature space.The diffusion map uses ϵ = 0.03.
Comparison to PCA and kernel PCA
The comparison shows why connectivity-based diffusion maps are preferable to PCA-based projections for topological classification of raw configurations. PCA and kernel PCA can fail when distortions erase sector separation or when kernel selection is inappropriate.
- Comparison to PCA and kernel PCA: The diffusion-map spectrum retains degenerate modes associated with topological sectors, while deviations from degeneracy scale exponentially with ϵ.The analysis derives the corresponding eigenstates and shows robustness to sample imbalance under stated assumptions.
- Comparison to PCA and kernel PCA: Different-winding samples can be closer than same-winding samples in PCA feature space, causing four-dimensional k-means to fail.Diffusion maps instead use connectivity to learn winding number directly from raw data.
- Comparison to PCA and kernel PCA: Kernel PCA performs better on the simple data set with a polynomial kernel, but its success depends on choosing an appropriate kernel.The polynomial kernel is d(x, y) = (1 + x · y)^d.
Clustering for random winding numbers and uneven sampling
Diffusion maps recover topological sectors despite random winding numbers and uneven sampling. A three-fold λk = 1 degeneracy and the leading nontrivial eigenvectors identify the sectors and assign their labels.
- Clustering across random winding numbers and uneven sampling: Three-fold degeneracy at λk = 1 signals three topological sectors in the unevenly sampled data.The sectors contain 500, 1000, and 2000 samples, respectively.
- Clustering across random winding numbers and uneven sampling: The first two nontrivial eigenvectors cluster samples into three topological sectors.The samples use winding-number components restricted to |νx,y| ≤ 3.
- Clustering across random winding numbers and uneven sampling: The clustering algorithm assigns the correct labels to the identified sectors.Samples are generated at T/J = 0.45, approximately Tc/2.
Diffusion maps as a function of system size
As temperature increases, diffusion-map clusters merge and cease to define topological sectors near the BKT transition. Across system sizes and kernel scales, the transition is estimated near Tc/J = 0.9 ± 0.1.
- Diffusion maps as a function of system size: For T/J < 0.8, sector classification is independent of system size, whereas for T/J > 1 the algorithm finds one cluster.The criterion is based on the relative quantities ¯σ and ¯D.
- Clustering across the BKT transition: For T/J ≤ 0.6, samples are tightly clustered by topological sector, while clusters begin merging at T/J = 0.7−0.8.For T/J ≥ 0.9, all clusters merge into one.
- Clustering across the BKT transition: For T/J ≥ 0.9, topological sectors are no longer well defined, marking the defining feature of the topological phase transition.The same qualitative behavior is observed for system lengths L = 48 and L = 64.
- Diffusion maps as a function of system size: Tc/J = 0.9 ± 0.1 is estimated over a wide range of diffusion-map kernel values.The transition estimate varies only within ±15% when ϵ changes by one order of magnitude.