Hopf Term for a Two-Dimensional Electron Gas: Comment to Reply to Comment

TL;DR

This paper discusses the debate over the derivation of the Hopf term in the hydrodynamic action for Skyrmion dynamics in quantum Hall effects, highlighting issues with the use of Euler angles.

Contribution

It clarifies the validity of the derivation of the Hopf term and addresses concerns about the use of ill-defined Euler angles in the previous models.

Findings

The original derivation of the Hopf term is confirmed to be problematic due to ill-defined variables.

The debate emphasizes the importance of proper variable choice in topological field theories.

The reply supports the original derivation's validity despite earlier concerns.

Abstract

In our Comment (G.E. Volovik and V.M. Yakovenko, Phys. Rev. Lett. 79 (1997) 3792) to the Letter by W. Apel and Yu.A. Bychkov (Phys. Rev. Lett. 78 (1997) 2188) we questioned the derivation of the Hopf term in the hydrodynamic action for the Skyrmion dynamics in QHE. Our main argument was that the description in terms of Euler angles used by Apel and Bychkov was potentially dangerous since these angles were ill-defined. The Reply (W. Apel and Yu.A. Bychkov, Phys. Rev. Lett. 79 (1997) 3792) to our Comment confirmed our apprehension.