Singular Behavior At The Edge of Laughlin States

TL;DR

This paper investigates the singular behavior of electron density at the edges of Laughlin fractional quantum Hall states, revealing universal features, edge dipole moments, and connections to bulk properties like odd viscosity.

Contribution

It introduces a perturbative method to compute edge density behavior, capturing overshoot phenomena and deriving edge dipole moments for Laughlin and Pfaffian states.

Findings

Edge density exhibits a universal overshoot near the boundary.

The perturbative expansion accurately models the dipole moment and overshoot.

Edge dipole moments relate to bulk odd viscosity.

Abstract

A distinguishing feature of fractional quantum Hall (FQH) states is a singular behavior of equilibrium densities at boundaries. In contrast to states at integer filling fraction, such quantum liquids posses an additional dipole moment localized near edges. It enters observable quantities such as universal dispersion of edge states and Lorentz shear stress. For a Laughlin state, this behavior is seen as a peak, or overshoot, in the single particle density near the edge, reflecting a general tendency of electrons in FQH states to cluster near edges. We compute the singular edge behavior of the one particle density by a perturbative expansion carried out around a completely filled Landau level. This correction is shown to fully capture the dipole moment and the major features of the overshoot observed numerically. Furthermore, it exhibits the Stokes phenomenon with the Stokes line at the boundary of the droplet, decaying like a Gaussian inside and outside the liquid with different decay lengths. In the limit of vanishing magnetic length the shape the overshoot is a singular double layer with a capacity that is a universal function of the filling fraction. Finally, we derive the edge dipole moment of Pfaffian FQH states. The result suggests an explicit connection between the magnitude of the dipole moment and the bulk odd viscosity.