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Nonlinear hydrodynamic response of a quantum Hall system

Hiroki Isobe

Abstract

The quantum Hall effect realizes a quantized Hall resistance $R_{xy} = h/(νe^2)$ whereas the longitudinal resistance vanishes. The quantized value consists of the fundamental physical quantities, the elementary charge $e$ and the Planck constant $h$, along with an integer or fractional constant $ν$. High precision measurements of $R_{xy}$ allude to a linear relation between the applied current $I$ and the Hall voltage $V_\mathrm{H}$. Here, we argue that a nonlinear relation between $I$ and $V_\mathrm{H}$ could arise when the electric field is spatially inhomogeneous. We first discuss that the linear $I$-$V_\mathrm{H}$ relation holds with Galilean invariance. Then we consider a hydrodynamic description of a quantum Hall liquid to deal with an axially symmetric electric field. It reveals a nonlinear electronic response arising from the centrifugal force exerted on a curved flow and the density gradient invoked by vorticity.

Nonlinear hydrodynamic response of a quantum Hall system

Abstract

The quantum Hall effect realizes a quantized Hall resistance whereas the longitudinal resistance vanishes. The quantized value consists of the fundamental physical quantities, the elementary charge and the Planck constant , along with an integer or fractional constant . High precision measurements of allude to a linear relation between the applied current and the Hall voltage . Here, we argue that a nonlinear relation between and could arise when the electric field is spatially inhomogeneous. We first discuss that the linear - relation holds with Galilean invariance. Then we consider a hydrodynamic description of a quantum Hall liquid to deal with an axially symmetric electric field. It reveals a nonlinear electronic response arising from the centrifugal force exerted on a curved flow and the density gradient invoked by vorticity.

Paper Structure

This paper contains 14 sections, 79 equations, 3 figures.

Table of Contents

  1. Lorentz invariance of a quantum Hall state
  2. Properties of quantum Hall fluids
  3. Incompressibility
  4. Stream function
  5. Dynamics of the vorticity
  6. Bernoulli's law
  7. The dimensionless parameter $\bm{\chi}$
  8. Integer quantum Hall state
  9. Fractional quantum Hall effect
  10. Quantum Hall fluid in a straight channel
  11. Quantum Hall fluid in axially symmetric systems
  12. Electric field proportional to $\bm{1/r}$
  13. Electric field proportional to $\bm{r}$
  14. Discussions

Figures (3)

  • Figure 1: Quantum Hall system in the Corbino geometry (A) Laughlin's setup to discuss the quantization of the Hall conductance. The magnetic field $\bm{B}$ points out of the plane. (B) Quantum Hall fluid in the Corbino geometry with the radial electric field. The Hall current flows in the azimuthal direction.
  • Figure 2: Geometry effects for quantum Hall fluids (A) Quantum Hall channels with opposite curvatures. $+$ and $-$ indicate the charge distribution for the flow to bend. (B) Classical trajectory of a cyclotron motion with a drift. (C) Quantum Hall fluid flow in a Mach--Zehnder interferometer. (D) Quantum Hall channel with the rectangular notches.
  • Figure S1: Cylindrical capacitor (A) Concentric cylindrical capacitor. $\lambda$ is the charge density per height and azimuthal angle. (B) Radial electric field inside the capacitor. $\varepsilon$ is the permittivity inside the capacitor.