Uncertainty Principle and Off-Diagonal Long Range Order in the Fractional Quantum Hall Effect

TL;DR

This paper extends the uncertainty principle to non-Hermitian operators and applies it to the fractional quantum Hall effect to demonstrate the absence of Bose-Einstein condensation and off-diagonal long-range order.

Contribution

It introduces a generalized uncertainty principle inequality for non-Hermitian operators and uses it to analyze long-range order in the FQHE, providing new theoretical insights.

Findings

Proves the absence of Bose-Einstein condensation in the FQHE wave function.

Links the lack of off-diagonal long-range order to the $q^2$ behavior of the static structure function.

Demonstrates the generalized uncertainty principle's application to 2D quantum systems.

Abstract

A natural generalization of the Heisenberg uncertainty principle inequality holding for non hermitian operators is presented and applied to the fractional quantum Hall effect (FQHE). This inequality was used in a previous paper to prove the absence of long range order in the ground state of several 1D systems with continuous group symmetries. In this letter we use it to rule out the occurrence of Bose-Einstein condensation in the bosonic representation of the FQHE wave function proposed by Girvin and MacDonald. We show that the absence of off-diagonal long range order in this 2D problem is directly connected with the $q^2$ behavior of the static structure function $S(q)$ at small momenta.