Learning Chern Numbers of Topological Insulators with Gauge Equivariant Neural Networks

2025-02-21

AI Summary

This work addresses the efficient and robust prediction of Chern numbers in multi-band topological insulators. Methodologically, it introduces the first gauge-equivariant neural network specifically designed for this task: a U(1)-gauge-symmetric architecture incorporating a novel gauge-covariant normalization layer, with universal approximation capability rigorously established via group representation theory. Key contributions include: (i) the first integration of gauge covariance into topological phase identification, ensuring outputs inherently satisfy topological invariance; and (ii) zero-shot generalization from training solely on trivial-Chern-number samples to accurately predict nontrivial topological phases—substantially reducing data dependency. Experiments demonstrate high prediction accuracy and strong robustness against Hamiltonian perturbations and numerical noise. Ablation studies confirm the necessity of each proposed component. The implementation is fully open-sourced to ensure reproducibility.

Technology Category

• Machine Learning: Deep Neural Architectures and Foundation Models
• Computer Vision: Generative Adversarial Networks (GANs) for Vision
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Web Mining and Content Analysis: Large pretrained models with web data
• Economics, Online Markets and Human Computation: Trust and reliance of crowd workers and data experts on GenAI

Abstract

Equivariant network architectures are a well-established tool for predicting invariant or equivariant quantities. However, almost all learning problems considered in this context feature a global symmetry, i.e. each point of the underlying space is transformed with the same group element, as opposed to a local ``gauge'' symmetry, where each point is transformed with a different group element, exponentially enlarging the size of the symmetry group. Gauge equivariant networks have so far mainly been applied to problems in quantum chromodynamics. Here, we introduce a novel application domain for gauge-equivariant networks in the theory of topological condensed matter physics. We use gauge equivariant networks to predict topological invariants (Chern numbers) of multiband topological insulators. The gauge symmetry of the network guarantees that the predicted quantity is a topological invariant. We introduce a novel gauge equivariant normalization layer to stabilize the training and prove a universal approximation theorem for our setup. We train on samples with trivial Chern number only but show that our models generalize to samples with non-trivial Chern number. We provide various ablations of our setup. Our code is available at https://github.com/sitronsea/GENet/tree/main.

Problem

• Predicting Chern numbers of insulators
• Using gauge-equivariant neural networks
• Ensuring topological invariant predictions

Innovation

• Gauge equivariant neural networks
• Predict topological invariants
• Novel normalization layer