Abstract
Non-Hermitian nodal knot metals (NKMs) contain intricate complex-valued energy bands which give rise to knotted exceptional loops and new topological surface states. We introduce a formalism that connects the algebraic, geometric, and topological aspects of these surface states with their parent knots. We also provide an optimized constructive ansatz for tight-binding models for non-Hermitian NKMs of arbitrary knot complexity and minimal hybridization range. Specifically, various representative non-Hermitian torus knots Hamiltonians are constructed in real-space, and their nodal topologies studied via winding numbers that avoid the explicit construction of generalized Brillouin zones. In particular, we identify the surface state boundaries as “tidal” intersections of the complex band structure in a marine landscape analogy. Beyond topological quantities based on Berry phases, we further find these tidal surface states to be intimately connected to the band vorticity and the layer structure of their dual Seifert surface, and as such provide a fingerprint for non-Hermitian NKMs.
Highlights
Non-Hermitian nodal knot metals (NKMs) contain intricate complex-valued energy bands which give rise to knotted exceptional loops and new topological surface states
We devise a comprehensive formalism that relates surface states of non-Hermitian NKMs to their Seifert surface topology, complex geometry, vorticity, and other bulk properties. Each of these properties has separately aroused much interest: Knot topology concerns the innumerable distinct configurations of knots, so intricate that they cannot be unambiguously classified by any single topological invariant; the complex-analytic structure of band structures have led to various non-Hermitian symmetry classifications which are augmented by the nonHermitian skin effect; and half-integer vorticity underscores the double-valuedness around exceptional points
A (p, q)-torus knot is one that winds p times around the symmetry axis while winding q times around the internal circle direction. These knots are isomorphic to closed braids with p strands each twisting q times around a torus, with the number of linked loops being the greatest common divisor (GCD) of p and q, i.e. GCD(p, q) linked loops[61,67,68,69], and encompasses many common knots like the Hopf-link and the Trefoil knot
Summary
Non-Hermitian nodal knot metals (NKMs) contain intricate complex-valued energy bands which give rise to knotted exceptional loops and new topological surface states. No systematic understanding of the shape, location, and topology of nonHermitian NKM surface state regions currently exists beyond isolated numerical results[34,41,42] This conceptual gap has endured until today, because non-Hermiticity modifies the topological bulk-boundary correspondence in subtle complexanalytic ways, which so far have not been studied beyond 1D, especially for models that possess complicated sets of hoppings across various distances[43,44,45,46,47,48,49,50,51]. We illustrate how the topological tidal surface states can be mapped out as topolectrical resonances in non-Hermitian circuit realizations, based on recent experimental demonstrations involving analogous 1D circuit arrays[62]
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
