Combinatorics

Physics

Topology (electrical circuits)

Pure mathematics

Knot (papermaking)

Mathematics

Mathematical physics

Hermitian matrix

Knot theory

Engineering

Ansatz

Trefoil knot

Surface (topology)

Geometry

Theoretical physics

Homogeneous space

Chemical engineering

Communications Physics

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.

Tidal surface states as fingerprints of non-Hermitian nodal knot metals