Machine Learning of Knot Topology in Non-Hermitian Band Braids

Abstract

The deep connection among braids, knots and topological physics has provided valuable insights into studying topological states in various physical systems. However, identifying distinct braid groups and knot topology embedded in non-Hermitian systems is challenging and requires significant efforts. Here, we demonstrate that an unsupervised learning with the representation basis of $su(n)$ Lie algebra on $n$-fold extended non-Hermitian bands can fully classify braid group and knot topology therein, without requiring any prior mathematical knowledge or any pre-defined topological invariants. We demonstrate that the approach successfully identifies different topological elements, such as unlink, unknot, Hopf link, Solomon ring, trefoil, and so on, by employing generalized Gell-Mann matrices in non-Hermitian models with $n$=2 and $n$=3 energy bands. Moreover, since eigenstate information of non-Hermitian bands is incorporated in addition to eigenvalues, the approach distinguishes the different parity-time symmetry and breaking phases, recognizes the opposite chirality of braids and knots, and identifies out distinct topological phases that were overlooked before. Our study shows significant potential of machine learning in classification of knots, braid groups, and non-Hermitian topological phases.

References in corpus (9)

• Observation of non-Abelian topological semimetals and their phase transitions
• Observation of Weyl point pair annihilation in a gyromagnetic photonic crystal
• Unsupervised Learning of Frustrated Classical Spin Models I: Principle Component Analysis
• Observation of Acoustic Non-Hermitian Bloch Braids and Associated Topological Phase Transitions
• Measuring the knot of non-Hermitian degeneracies and non-commuting braids
• Probing Complex-energy Topology via Non-Hermitian Absorption Spectroscopy in a Trapped Ion Simulator
• Minimal non-abelian nodal braiding in ideal metamaterials
• Unsupervised learning of topological phase diagram using topological data analysis
• Gain-loss-induced non-Abelian Bloch braids

Cited by in corpus (6)

• Quantum entanglement and non-Hermiticity in free-fermion systems
• Braiding Topology of Non-Hermitian Open-Boundary Bands
• Versatile Braiding of Non-Hermitian Topological Edge States
• Skin Effect Induced Anomalous Dynamics from Charge-Fluctuating Initial States
• Machine learning topological energy braiding of non-Bloch bands
• Emergence of Nodal-Knot Transitions by Disorder