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

## TL;DR

This paper introduces a formalism linking the complex surface states of non-Hermitian nodal knot metals to their underlying knots, providing models and topological insights that serve as fingerprints for these exotic materials.

## Contribution

It develops a formalism connecting algebraic, geometric, and topological aspects of surface states with knots, and constructs minimal tight-binding models for non-Hermitian NKMs of arbitrary complexity.

## Key findings

- Constructed real-space Hamiltonians for non-Hermitian torus knots.
- Identified surface state boundaries as 'tidal' intersections in complex band structures.
- Linked tidal surface states to band vorticity and Seifert surface topology.

## Abstract

Non-Hermitian nodal knot metals (NKMs) contains intricate complex-valued energy bands gives 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, and 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.