Global symmetries of quantum Hall systems: lattice description

TL;DR

This paper explores the non-local symmetries in lattice Chern-Simons models relevant to quantum Hall systems, revealing how dualities interchange magnetic flux and current, and how flux attachment modifies flux, providing insights into phase diagram symmetries and current-voltage relations.

Contribution

It demonstrates that particle-vortex duality and flux attachment transformations act as symmetries in lattice models of quantum Hall systems, connecting to observed phase diagram behaviors.

Findings

Particle-vortex duality interchanges flux and current.

Flux attachment increases flux by current component.

Symmetries correspond to phase diagram laws and current-voltage mappings.

Abstract

I analyze non-local symmetries of finite-size Euclidean 3D lattice Chern-Simons models in the presence of an external magnetic field and non-zero average current. It is shown that under very general assumptions the particle-vortex duality interchanges the total Euclidean magnetic flux Phi/(2 Pi) and the total current I in a given direction, while the flux attachment transformation increases the flux in a given direction by the corresponding component of the current, Phi -> Phi+4 Pi I, independently of the disorder. In the language of 2+1 dimensional models, appropriate for describing quantum Hall systems, these transformations are equivalent to the symmetries of the phase diagram known as the Correspondence Laws, and the non-linear current-voltage mapping between mutually dual points, recently observed near the quantum Hall liquid-insulator transitions.