Hyperbolic Nodal Band Structures and Knot Invariants

Abstract

We extend the list of known band structure topologies to include a large family of hyperbolic nodal links and knots, occurring both in conventional Hermitian systems where their stability relies on discrete symmetries, and in the dissipative non-Hermitian realm where the knotted nodal lines are generic and thus stable towards any small perturbation. We show that these nodal structures, taking the forms of Turk's head knots, appear in both continuum- and lattice models with relatively short-ranged hopping that is within experimental reach. To determine the topology of the nodal structures, we devise an efficient algorithm for computing the Alexander polynomial, linking numbers and higher order Milnor invariants based on an approximate and well controlled parameterisation of the knot.

23 pages; New subsection (2.3) added with more explicit examples, 2 new figures, additional references and typos corrected

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Cited by in corpus (19)

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• Topological Non-Hermitian skin effect
• Symmetry and Higher-Order Exceptional Points
• Knots and Non-Hermitian Bloch Bands
• Bulk-boundary correspondence in non-Hermitian systems: stability analysis for generalized boundary conditions
• Knot topology of exceptional point and non-Hermitian no-go theorem
• Exceptional Non-Abelian Topology in Multiband Non-Hermitian Systems
• Boundary states of 4D topological matter: Emergence and full 3D-imaging of nodal Seifert surfaces
• Phase Transitions and Generalized Biorthogonal Polarization in Non-Hermitian Systems
• Classification of Exceptional Nodal Topologies Protected by $\mathcal{P}\mathcal{T}$ Symmetry
• Homotopy, Symmetry, and Non-Hermitian Band Topology
• Symmetry-protected exceptional and nodal points in non-Hermitian systems
• Enhanced higher harmonic generation from nodal topology
• Degeneracy and defectiveness in non-Hermitian systems with open boundary
• Braiding Topology of Non-Hermitian Open-Boundary Bands
• Quantum tomography of a third-order exceptional point in a dissipative trapped ion
• $\mathcal{P}\mathcal{T}$ symmetry-protected exceptional cones and analogue Hawking radiation
• Fractal Nodal Band Structures
• The analytically tractable zoo of similarity-induced exceptional structures