Quantum Mechanics on Thin Cylinders

TL;DR

This paper explores quantum particles with various statistics on an infinite cylinder, analyzing effects of magnetic fields, interactions, and potential experimental observations, and mapping complex systems to simpler models like Ising and Sutherland models.

Contribution

It introduces mappings of particles on a cylinder to effective one-dimensional models, including Ising and Sutherland models, under magnetic fields and interactions.

Findings

Magnetic field breaks translational symmetry via Aharonov-Bohm effect.

Interacting fermions map to an effective 1D lattice or Ising chain.

Anyons relate to the Sutherland model in certain limits.

Abstract

We discuss the quantum mechanics of particles of arbitrary statistics on an infinite cylinder with and without a magnetic field perpendicular to the surface. In the presence of a magnetic field, the translational symmetry along the cylinder is broken down to a discrete one by the Aharonov-Bohm effect. For interacting fermions in a strong field we get an effective one-dimensional lattice model that in a limit can be mapped on an Ising chain. We also show that a system of anyons on a cylinder are, in a certain limit closely related to the one-dimensional (integrable) Sutherland model. By order of magnitude estimates we demonstrate that none of these effects are likely to be experimentally observed with present techniques.