Localization on a synthetic Hall cylinder

TL;DR

This paper demonstrates that transforming a synthetic Hall ribbon into a Hall cylinder introduces an intrinsic lattice, leading to localization phenomena when combined with external potentials, and explores flux dependence near localization transitions.

Contribution

It introduces the concept of localization on a synthetic Hall cylinder caused by incommensurate lattices and investigates flux effects on physical observables near transition points.

Findings

Localization occurs on a synthetic Hall cylinder with incommensurate lattices.

Physical observables depend on axial magnetic flux near transition points.

In the irrational limit, flux fluctuations do not affect observables.

Abstract

By engineering laser-atom interactions, both Hall ribbons and Hall cylinders as fundamental theoretical tools in condensed matter physics have recently been synthesized in laboratories. Here, we show that turning a synthetic Hall ribbon into a synthetic Hall cylinder could naturally lead to localization. Unlike a Hall ribbon, a Hall cylinder hosts an intrinsic lattice, which arises due to the periodic boundary condition in the azimuthal direction, in addition to the external periodic potential imposed by extra lasers. When these two lattices are incommensurate, localization may occur on a synthetic Hall cylinder. Near the localization-delocalization transitions, the dependence of physical observables on the axial magnetic flux allows us to tackle a fundamental question of determining the accuracy of rational approximation of irrational numbers. In the irrational limit, physical observables are no longer affected by fluctuations of the axial flux.