Cyclotron orbit knot and tunable-field quantum Hall effect

The Bohr-Sommerfeld quantization of the cyclotron orbit in a 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 structure. We present a scenario of a Weyl semimetal slab, where the Fermi arcs on the opposing surfaces can cross without interfering with each other and form a knot together with the bulk chiral Landau levels. We provide a microscopic lattice model with cyclotron orbits of trefoil-knot geometry 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.