Knots and Non-Hermitian Bloch Bands

Abstract

Knots have a twisted history in quantum physics. They were abandoned as failed models of atoms. Only much later was the connection between knot invariants and Wilson loops in topological quantum field theory discovered. Here we show that knots tied by the eigenenergy strings provide a complete topological classification of one-dimensional non-Hermitian (NH) Hamiltonians with separable bands. A ${\mathbb{Z}}_2$ knot invariant, the global biorthogonal Berry phase $Q$ as the sum of the Wilson loop eigenphases, is proved to be equal to the permutation parity of the NH bands. We show the transition between two phases characterized by distinct knots occur through exceptional points and come in two types. We further develop an algorithm to construct the corresponding tight-binding NH Hamiltonian for any desired knot, and propose a scheme to probe the knot structure via quantum quench. The theory and algorithm are demonstrated by model Hamiltonians that feature for example the Hopf link, the trefoil knot, the figure-8 knot and the Whitehead link.

8+10 pages, 3+4 figures

References in corpus (13)

• Classification of topological insulators and superconductors in three spatial dimensions
• Making Sense of Non-Hermitian Hamiltonians
• The physics of exceptional points
• Topological Origin of Non-Hermitian Skin Effects
• Loss-induced suppression and revival of lasing
• PT-Symmetric Phonon Laser
• Selective enhancement of topologically induced interface states in a dielectric resonator chain
• Weyl Exceptional Rings in a Three-Dimensional Dissipative Cold Atomic Gas
• Periodic Table for Topological Bands with Non-Hermitian Bernard-LeClair Symmetries
• New topological invariants in non-Hermitian systems
• Topological invariance and global Berry phase in non-Hermitian systems
• Detecting topological invariants in nonunitary discrete-time quantum walks
• Three-dimensional Pentagon Carbon with a genesis of emergent fermions

Cited by in corpus (102)

• Non-Hermitian Topology and Exceptional-Point Geometries
• Topological Non-Hermitian skin effect
• Observation of Acoustic Non-Hermitian Bloch Braids and Associated Topological Phase Transitions
• Measuring the knot of non-Hermitian degeneracies and non-commuting braids
• Geometric Response and Disclination-Induced Skin Effects in Non-Hermitian Systems
• Non-Hermitian skin effect and self-acceleration
• Knot topology of exceptional point and non-Hermitian no-go theorem
• Enhancement of Second-Order Non-Hermitian Skin Effect by Magnetic Fields
• Non-Hermitian Topological Phases: Principles and Prospects
• Exceptional Non-Abelian Topology in Multiband Non-Hermitian Systems
• Non-Hermitian Photonic Lattices: tutorial
• Probing Complex-energy Topology via Non-Hermitian Absorption Spectroscopy in a Trapped Ion Simulator
• Eigenvalue topology of non-Hermitian band structures in two and three dimensions
• Topological origin of non-Hermitian skin effect in higher dimensions and uniform spectra
• Symmetry and topological classification of Floquet non-Hermitian systems
• Homotopy, Symmetry, and Non-Hermitian Band Topology
• Theoretical Prediction of Non-Hermitian Skin Effect in Ultracold Atom Systems
• Synthetic non-Abelian gauge fields for non-Hermitian systems
• Braid Protected Topological Band Structures with Unpaired Exceptional Points
• Topological energy braiding of the non-Bloch bands
• Non-Hermitian Maryland Model
• Floquet Non-Abelian Topological Insulator and Multifold Bulk-Edge Correspondence
• Symmetry-protected exceptional and nodal points in non-Hermitian systems
• Topology of multipartite non-Hermitian one-dimensional systems
• Manipulating non-Hermitian skin effect via electric fields
• Non-Hermitian spatial symmetries and their stabilized normal and exceptional topological semimetals
• Anatomy of open-boundary bulk in multiband non-Hermitian systems
• Non-Hermitian Weyl semimetal and its Floquet engineering
• Degeneracy and defectiveness in non-Hermitian systems with open boundary
• Phase transitions and bunching of correlated particles in a non-Hermitian quasicrystal
• Experimental realization of stable exceptional chains protected by non-Hermitian latent symmetries unique to mechanical systems
• Non-Hermitian eigenvalue knots and their isotopic equivalence
• Non-Abelian Braiding of Topological Edge Bands
• Modified Generalized-Brillouin-Zone Theory with On-site Disorders
• Unsupervised Learning of Topological Non-Abelian Braiding in Non-Hermitian Bands
• Real spectra, Anderson localization, and topological phases in one-dimensional quasireciprocal systems
• Machine Learning of Knot Topology in Non-Hermitian Band Braids
• Characterizing and Tuning Exceptional Points Using Newton Polygons
• Braiding Topology of Non-Hermitian Open-Boundary Bands
• Spectral structure and doublon dissociation in the two-particle non-Hermitian Hubbard model
• Topological transitions in quantum jump dynamics: Hidden exceptional points
• Evolution of spectral topology in one-dimensional long-range nonreciprocal lattices
• Dynamical signatures of point-gap Weyl semimetal
• Unravelling the edge spectra of non-Hermitian Chern insulators
• Anomalous dynamical response of non-Hermitian topological phases
• Nontrivial worldline winding in non-Hermitian quantum systems
• Hermitian Topologies originating from non-Hermitian braidings
• Selective and tunable excitation of topological non-Hermitian skin modes
• Electromagnetic Proximity Effect: Superconducting Magnonics and Beyond
• Versatile Braiding of Non-Hermitian Topological Edge States
• Constraints of internal symmetry on the non-Hermitian skin effect and bidirectional skin effect under the action of the Hermitian conjugate of time-reversal symmetry
• Investigation of a non-Hermitian edge burst with time-dependent perturbation theory
• Quantum tomography of a third-order exceptional point in a dissipative trapped ion
• Classification of multiple arbitrary-order non-Hermitian singularities
• Anomalous second-order skin modes in Floquet non-Hermitian systems
• Extended imaginary gauge transformation in a general nonreciprocal lattice
• Skin Effect Induced Anomalous Dynamics from Charge-Fluctuating Initial States
• General theory for infernal points in non-Hermitian systems
• Braids and Higher-order Exceptional Points from the Interplay Between Lossy Defects and Topological Boundary States
• Electric polarization and its quantization in one-dimensional non-Hermitian chains
• Implementing non-Abelian Hatano-Nelson model in electric circuits
• Inner non-Hermitian skin effect on Bethe lattice
• Gain-loss-induced non-Abelian Bloch braids
• Topological Invariant for Multi-Band Non-hermitian Systems with Chiral Symmetry
• One-dimensional non-Hermitian band structures as Riemann surfaces
• Spin-dependent gain and loss in photonic quantum spin Hall systems
• Measuring Non-Hermitian Topological Invariants Directly from Quench Dynamics
• Realization of geometric phase topology induced by multiple exceptional points
• Theory for the spectral splitting exponent of exceptional points
• Topological eigenvalues braiding and quantum state transfer near a third-order exceptional point
• Tentaclelike spectra and bound states in Hatano-Nelson chain with long-range impurity coupling
• Contactless generation and trapping of hydrodynamic knots in sessile droplets by acoustic screw dislocations
• Observation of Braid-Protected Unpaired Exceptional Points
• Topological Dynamics and Correspondences in Composite Exceptional Rings
• Pseudo-Hermitian Topology of Multiband Non-Hermitian Systems
• Recent advances in the combination of nonlinearity and exceptional points
• Non-Hermitian Delocalization Induced by Residue Imaginary Velocity
• Exceptional Non-Hermitian Topology Associated with Non-Toroidal Brillouin Zones
• Non-Abelian geometry, topology, and dynamics of a nonreciprocal Su-Schrieffer-Heeger ladder
• Non-Hermitian Hopf insulators
• Measuring topological invariants of even-dimensional line-gapped non-Hermitian systems through quench dynamics
• Nodal Spectral Functions Stabilized by Non-Hermitian Topology of Quasiparticles
• Machine learning topological energy braiding of non-Bloch bands
• Experimental Realization of Non-Abelian Permutations in a Three-State Non-Hermitian System
• Non-Abelian interference of topological edge states
• Entanglement manifestation of knot topology in a non-Hermitian lattice
• Higher-order Topological Knots and the classification of non-Hermitian lattices under $C_n$ symmetry
• Self-healing of non-Hermitian topological skin modes
• Emergence of Nodal-Knot Transitions by Disorder
• 1D Spontaneous Symmetry Breaking in thermal equilibrium via Non-Hermitian Construction
• Exceptional topology on nonorientable manifolds
• Emergence of Hermitian topology from non-Hermitian knots
• Non-orientable Exceptional Points in Twisted Boundary Systems
• Winding-control mechanism of non-Hermitian systems
• Anatomy of Non-Hermitian Dynamical Quantum Phase Transitions
• Braiding of dynamical eigenvalues of Hermitian bosonic Kitaev chains
• Topology of the generalized Brillouin zone of one-dimensional models
• Wiener-Hopf factorization and non-Hermitian topology for Amoeba formulation in one-dimensional multiband systems
• Survival of Hermitian Criticality in the Non-Hermitian Framework
• Coexistence of topological Anderson insulator and multifractal critical phase in a non-Hermitian quasicrystal
• Anomalous Transport Induced by Non-Hermitian Anomalous Berry Connection in Non-Hermitian Systems
• Illuminating the bulk-boundary correspondence of a non-Hermitian stub lattice with Majorana stars