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

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.

arXiv comment10 pages, 7 figures

References in corpus (58)

• Topological Insulators
• Topological insulators and superconductors
• Classification of topological quantum matter with symmetries
• Solving the Quantum Many-Body Problem with Artificial Neural Networks
• Periodic table for topological insulators and superconductors
• Floquet Topological Insulator in Semiconductor Quantum Wells
• Machine learning phases of matter
• Topological phases of non-Hermitian systems
• Symmetry and Topology in Non-Hermitian Physics
• Topological characterization of periodically-driven quantum systems
• Realizing Majorana zero modes in superconductor-semiconductor heterostructures
• Topological phases of fermions in one dimension
• Many-body quantum state tomography with neural networks
• Learning phase transitions by confusion
• Inversion Symmetric Topological Insulators
• Quantum Entanglement in Neural Network States
• Machine learning quantum phases of matter beyond the fermion sign problem
• Efficient Representation of Quantum Many-body States with Deep Neural Networks
• Restricted-Boltzmann-Machine Learning for Solving Strongly Correlated Quantum Systems
• Machine Learning Topological States
• Identifying topological order through unsupervised machine learning
• Quantum Loop Topography for Machine Learning
• Machine Learning Topological Invariants with Neural Networks
• Twisted equivariant matter
• Symmetries and many-body excited states with neural-network quantum states
• Identifying Quantum Phase Transitions using Artificial Neural Networks on Experimental Data
• A Neural Decoder for Topological Codes
• Machine learning vortices at the Kosterlitz-Thouless transition
• Unsupervised machine learning and band topology
• Optimizing Quantum Error Correction Codes with Reinforcement Learning
• Machine Learning for Quantum Matter
• Integrating Neural Networks with a Quantum Simulator for State Reconstruction
• Machine-learning-assisted correction of correlated qubit errors in a topological code
• Identifying Quantum Phase Transitions with Adversarial Neural Networks
• Machine Learning ${\mathbb{Z}}_2$ Quantum Spin Liquids with Quasi-particle Statistics
• Machine learning of phase transitions in the percolation and XY models
• Unsupervised Manifold Clustering of Topological Phononics
• Quantum error correction for the toric code using deep reinforcement learning
• Chiral Topological Phases from Artificial Neural Networks
• Global optimization of quantum dynamics with AlphaZero deep exploration
• Deep Learning Topological Invariants of Band Insulators
• Equivalent topological invariants for one-dimensional Majorana wires in symmetry class D
• Unsupervised identification of topological order using predictive models
• Real space mapping of topological invariants using artificial neural networks
• Hamiltonian Learning for Quantum Error Correction
• Automated discovery of characteristic features of phase transitions in many-body localization
• Machine learning topological phases in real space
• Interpreting machine learning of topological quantum phase transitions
• Interpretable machine learning for inferring the phase boundaries in a nonequilibrium system
• Machine learning topological invariants of non-Hermitian systems
• Generalized Transfer Matrix States from Artificial Neural Networks
• Unsupervised learning using topological data augmentation
• Quantum topology identification with deep neural networks and quantum walks
• Revealing quantum chaos with machine learning
• Deep learning of topological phase transitions from entanglement aspects
• Inversion symmetry protected topological insulators and superconductors
• Detecting Nematic Order in STM/STS Data with Artificial Intelligence
• Machine learning assisted measurement of local topological invariants

Cited by in corpus (5)

• Provably efficient machine learning for quantum many-body problems
• Machine learning topological invariants of non-Hermitian systems
• Hessian-based toolbox for reliable and interpretable machine learning in physics
• Characterizing out-of-distribution generalization of neural networks: application to the disordered Su-Schrieffer-Heeger model
• Quaternion-based machine learning on topological quantum systems