Coherent State Wave Functions on a Torus with a Constant Magnetic Field

TL;DR

This paper compares two definitions of localized states in the lowest Landau level on a torus, analyzing their properties and localization, and applies the findings to quantum Hall liquids.

Contribution

It introduces and compares two methods for defining localized states in the LLL on a torus, revealing that only the projected delta function achieves maximal localization.

Findings

Projected delta function is maximally localized.

Functions with zeros at a point are less localized.

Methods are applied to hierarchical quantum Hall liquids.

Abstract

We study two alternative definitions of localized states in the lowest Landau level (LLL) on a torus. One definition is to construct localized states, as projection of the coordinate delta function onto the LLL. Another definition, proposed by Haldane, is to consider the set of functions which have all their zeros at a single point. Since a LLL wave function on a torus, supporting $N_\phi$ magnetic flux quanta, is uniquely defined by the position of its $N_\phi$ zeros, this defines a set of functions that are expected to be localized around the point maximally far away form the zeros. These two families of localized states have many properties in common with the coherent states on the plane and on the sphere, {\em viz.} a resolution of unity and a self-reproducing kernel. However, we show that only the projected delta function is maximally localized. Additionally, we show how to project onto the LLL, functions that contain holomorphic derivatives and/or anti-holomorphic polynomials, and apply our methods in the description of hierarchical quantum Hall liquids