Time-dependent Magnetic Fields and the Quantum Hall Effect

TL;DR

This paper extends the analysis of time-dependent harmonic oscillators to the Landau problem, deriving generalized Laughlin wave functions and studying the dynamics of density fluctuations and edge modes in a time-varying magnetic field.

Contribution

It introduces a method to handle time-dependent magnetic fields in quantum Hall systems and constructs generalized wave functions for such scenarios.

Findings

Derived generalized Laughlin wave functions for time-dependent magnetic fields.

Analyzed the dynamics of density fluctuations and edge modes under time-varying magnetic fields.

Demonstrated the possibility of tuning magnetic field frequency to control fermionic droplet properties.

Abstract

Ermakov has shown how the solution to the classical harmonic oscillator in one spatial dimension with general time-dependent frequency can be reduced to the time-independent case and an associated nonlinear ordinary differential equation, an analysis which has been applied to the Schr\"odinger equation as well. We extend this analysis to the Landau problem of a charged particle in a uniform magnetic field in two dimensions and construct the generalized Laughlin wave functions for the case when the magnetic field is time-dependent. We also work out the dynamics of density fluctuations (the Girvin, MacDonald, Platzman or GMP mode) and argue that it is possible to tune the frequency of the magnetic field to obtain a compressible droplet of fermions. We also analyze the dynamics of the edge modes of the droplet for the integer Hall effect.