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Machine Learning of Topological Insulator and Anderson Insulator in One-Dimensional Extended Su-Schrieffer-Heeger Chain

Zhekai Yin, C. K. Ong

Abstract

We study disorder effects in the extended Su-Schrieffer-Heeger (SSH) model using a convolutional neural network (CNN) trained on reduced correlation matrices (RCMs) of disorder-free systems to predict winding number phase diagrams in systems with off-diagonal and diagonal disorder. The trained CNN model generalizes to chiral-symmetry-preserving off-diagonal disorder system but fails in the presence of chiral-symmetry-breaking diagonal disorder system. Using principal component analysis (PCA) of the RCM feature space, we demonstrate that disorder-free and symmetry-preserving systems share overlapping feature manifolds, whereas symmetry-breaking disorder causes them to diverge. Inverse participation ratio (IPR) and energy spectrum analysis further demonstrate that off-diagonal disorder preserves topological edge states, whereas diagonal disorder drives a transition to an Anderson insulator. Our results position machine learning not merely as a classifier, but as a sensitive probe for the symmetry-protected nature of quantum matter.

Machine Learning of Topological Insulator and Anderson Insulator in One-Dimensional Extended Su-Schrieffer-Heeger Chain

Abstract

We study disorder effects in the extended Su-Schrieffer-Heeger (SSH) model using a convolutional neural network (CNN) trained on reduced correlation matrices (RCMs) of disorder-free systems to predict winding number phase diagrams in systems with off-diagonal and diagonal disorder. The trained CNN model generalizes to chiral-symmetry-preserving off-diagonal disorder system but fails in the presence of chiral-symmetry-breaking diagonal disorder system. Using principal component analysis (PCA) of the RCM feature space, we demonstrate that disorder-free and symmetry-preserving systems share overlapping feature manifolds, whereas symmetry-breaking disorder causes them to diverge. Inverse participation ratio (IPR) and energy spectrum analysis further demonstrate that off-diagonal disorder preserves topological edge states, whereas diagonal disorder drives a transition to an Anderson insulator. Our results position machine learning not merely as a classifier, but as a sensitive probe for the symmetry-protected nature of quantum matter.
Paper Structure (18 sections, 7 equations, 9 figures)

This paper contains 18 sections, 7 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Su-Schrieffer-Heeger Model
  3. Hamiltonian
  4. Winding Number
  5. Symmetry and Stability Under Disorder
  6. Convolutional Neural Network (CNN)
  7. Input Data: Reduced Correlation Matrix (RCM)
  8. CNN Architecture
  9. CNN Predictions under Disorder
  10. OOD Prediction for Off-Diagonal Disorder Case
  11. OOD Prediction for Diagonal Disorder Case
  12. PCA Analysis of RCM Feature Space
  13. Edge States, Localization, and Symmetry Breaking
  14. IPR as Spectral Diagnostics
  15. Energy Spectrum of Disorder-free, Off-diagonal Disorder and Diagonal Disorder Cases
  16. ...and 3 more sections

Figures (9)

  • Figure 1: Schematic diagram of SSH model with long range hopping. $A_i,B_i$ are sites of the $i$-th unit cell. $\{t_{1,1},t_{1,2},t_{1,3},...\}$ are the intra-cell hopping amplitudes, $\{t_{2,1},t_{2,2},t_{2,3},...\}$ are the inter-cell hopping amplitudes, $t_3$ is the long range hopping amplitude, $\{V_1,V_2,V_3,...\}$ are the on-site potentials.
  • Figure 2: Schematic diagram of CNN architecture. The input is RCM of size $1 \times 40 \times 40$. The feature extraction stage consists of three convolutional layers (using $3 \times 3$ kernels) and two max-pooling layers (using $2 \times 2$ kernels) to progressively extract high-level topological features while reducing spatial dimensionality. The final feature maps of size $64 \times 10 \times 10$ are flattened into a 6400-dimensional feature vector. This vector is passed through two fully connected layers, reducing the dimensionality to 256 and finally to 3. A Softmax activation function is applied at the output layer to provide the predicted probabilities for winding numbers $\{0, 1, 2\}$.
  • Figure 3: Analytical winding number map of disorder-free system
  • Figure 4: Visualization results for $W_{\text{off}}=0.1$: (a) winding number $\nu$ map, boundary is the same as disorder-free system, (b) confidence map indicating prediction certainty, confidence is very high throughout the map, (c) entropy map representing prediction uncertainty, entropy is very low throughout the map.
  • Figure 5: Visualization results for $V_{\text{on}}=0.1$: (a) winding number $\nu$ map, no phase boundary can be observed, (b) confidence map indicating prediction certainty, confidence is very low throughout the map, (c) entropy map representing prediction uncertainty, entropy is very high throughout the map.
  • ...and 4 more figures