Asymptotically exact dispersion relations for collective modes in a confined charged Fermi liquid

TL;DR

This paper derives precise dispersion relations for edge modes in a confined electron liquid, incorporating effects of screening, pressure, and shear modulus, with applications to quantum dots in magnetic fields.

Contribution

It provides exact dispersion relations for edge modes in a confined Fermi liquid using local conservation laws, including effects of screening and shear modulus, and applies to quantum dots.

Findings

Dispersion relations are accurate up to λ²q² terms.

Expressions relate dispersion to static pressure and shear modulus.

Frequency shifts of Kohn modes in quantum dots are derived.

Abstract

Using general local conservations laws we derive dispersion relations for edge modes in a slab of electron liquid confined by a symmetric potential. The dispersion relations are exact up to $\lambda^{2} q^{2}$, where $q$ is a wave vector and $\lambda$ is an effective screening length. For a harmonic external potential the dispersion relations are expressed in terms of the {\em exact} static pressure and dynamic shear modulus of a homogeneous liquid with the density taken at the slab core. We also derive a simple expression for the frequency shift of the dipole (Kohn) modes in nearly parabolic quantum dots in a magnetic field.