Unsupervised interpretable learning of topological indices invariant under permutations of atomic bands

TL;DR

This paper introduces a neural network approach to identify topological indices in multi-band insulators that remain invariant under transformations, extending previous methods with a simplified data scheme and a specialized neural network layer.

Contribution

The work generalizes and simplifies data generation for topological data augmentation and introduces a neural network with a 'mod' layer for $Z_n$ classification, improving unsupervised topological index detection.

Findings

Successfully identifies topological indices invariant under atomic-limit transformations.

Extends topological data augmentation with a simplified scheme.

Develops an interpretable neural network capturing momentum space topological properties.

Abstract

Multi-band insulating Bloch Hamiltonians with internal or spatial symmetries, such as particle-hole or inversion, may have topologically disconnected sectors of trivial atomic-limit (momentum-independent) Hamiltonians. We present a neural-network-based protocol for finding topologically relevant indices that are invariant under transformations between such trivial atomic-limit Hamiltonians, thus corresponding to the standard classification of band insulators. The work extends the method of "topological data augmentation" for unsupervised learning introduced in Ref. [1] by also generalizing and simplifying the data generation scheme and by introducing a special "mod" layer of the neural network appropriate for $Z_n$ classification. Ensembles of training data are generated by deforming seed objects in a way that preserves a discrete representation of continuity. In order to focus the learning on the topologically relevant indices, prior to the deformation procedure we stack the seed Bloch Hamiltonians with a complete set of symmetry-respecting trivial atomic bands. The obtained datasets are then used for training an interpretable neural network specially designed to capture the topological properties by learning physically relevant momentum space quantities, even in crystalline symmetry classes.