# Cyclotron orbit knot and tunable-field quantum Hall effect

## TL;DR

This paper explores how knotted cyclotron orbits in Weyl semimetals influence quantum Hall effects, revealing novel topological phenomena and tunable magnetic responses due to the orbit's complex geometry.

## Contribution

It introduces a microscopic lattice model for trefoil knot cyclotron orbits in Weyl semimetals and analyzes their impact on quantum oscillations and Berry phase.

## Key findings

- Trefoil knot cyclotron orbits can be realized in Weyl semimetals.
- Knotting alters the Berry phase and quantization conditions.
- The magnetic field response becomes tunable due to the orbit's topology.

## Abstract

From a semiclassical perspective, the Bohr-Sommerfeld quantization of the closed cyclotron orbits for charged particles such as electrons in an external magnetic field gives rise to discrete Landau levels and a series of fascinating quantum Hall phenomena. Here, we consider topologically nontrivial physics from a distinct origin, where the cyclotron orbits take nontrivial knotting structures such as a trefoil knot. We present a scenario of a Weyl semimetal with a slab geometry, where the Fermi arcs on the opposing surfaces can cross without interfering with each other and form a knot together with the bulk Weyl nodes, and in an external magnetic field, the resulting chiral Landau levels. We provide a microscopic lattice model to realize a cyclotron orbit with an unconventional geometry of a trefoil knot and study the corresponding quantum oscillations. Interestingly, unlike the conventional ring-shaped cyclotron orbit, a trefoil knot is self-threading, allowing the magnetic field line along the cyclotron orbit to contribute to the overall Berry phase and therefore altering the external magnetic field for each quantization level. The cyclotron orbit knot offers an arena of the nontrivial knot theory in three spatial dimensions and its subsequent physical consequences.