Momentum Distribution Function of a Narrow Hall Bar in the FQHE Regime

TL;DR

This paper investigates the momentum distribution function of narrow Hall bars in the fractional quantum Hall regime, revealing additional singularities at smaller odd multiples of the Fermi wavevector and analyzing their dependence on width and interactions.

Contribution

It introduces the occurrence of new singularities in the momentum distribution for narrow Hall bars and characterizes their exponents and amplitudes, extending previous understanding from wide bars.

Findings

Additional singularities at odd multiples of $k_F$ in narrow bars.

Exponent $2 \Delta_{p}$ is width-independent if inter-edge interactions are neglected.

Amplitude of singularities vanishes exponentially with width for $p eq M$.

Abstract

The momentum distribution function ($n(k)$) of a narrow Hall bar in the fractional quantum Hall effect regime is investigated using Luttinger liquid and microscopic many-particle wavefunction approaches. For wide Hall bars with filling factor $\nu=1/M$, where $M$ is an odd integer, $n(k)$ has singularities at $\pm M k_F$. We find that for narrow Hall bars additional singularities occur at smaller odd integral multiples of $k_F$: $ n(k) \sim A_p \mid k\pm pk_{F} \mid ^{2\Delta_{p}-1}$ near $k=\pm pk_{F}$, where $p$ is an odd integer $M,M-2,M-4,...,1$. If inter-edge interactions can be neglected, the exponent $2 \Delta_{p}= (1/ \nu+p^{2} \nu)/2$ is independent of the width ($w$) of the Hall bar but the amplitude of the singularity $A_p$ vanishes exponentially with $w$ for $p\not=M$.