A note on GMP algebra, dipole symmetry, and Hohenberg-Mermin-Wagner theorem in the lowest Landau level

TL;DR

This paper explores how dipole symmetry in the lowest Landau level constrains symmetry breaking, and how inhomogeneous magnetic fields affect this symmetry and the associated algebra, extending understanding of quantum Hall systems.

Contribution

It demonstrates that dipole symmetry forbids spontaneous U(1) symmetry breaking at zero temperature and generalizes the GMP algebra to inhomogeneous magnetic fields.

Findings

Dipole symmetry prevents U(1) symmetry breaking at zero temperature.

In inhomogeneous magnetic fields, dipole symmetry is lost but the GMP algebra persists.

The generalized GMP algebra remains valid despite the absence of dipole symmetry.

Abstract

After projection to the lowest Landau level translational invariance and particle conservation combine into dipole symmetry. We show that the new symmetry forbids spontaneous $U(1)$ symmetry breaking at zero temperature. In the case of the spatially inhomogeneous magnetic field, where the translational invariance is absent, we show that the dipole symmetry disappears and the constraint on the symmetry breaking is lifted. We pay special attention to the fate of the Girvin-Macdonald-Platzman algebra in the inhomogeneous magnetic field and show that a natural generalization of it is still present even though the dipole symmetry is not.