Wave Topology in Hall MHD

TL;DR

This paper derives the complete wave spectrum of Hall MHD, revealing its topological structure and showing it is homotopic to ideal MHD without additional wave branches.

Contribution

It provides the first comprehensive eigenmode spectrum and topological analysis of HMHD waves, clarifying their relation to ideal MHD.

Findings

HMHD wave spectrum consists of three branches: slow, shear Alfvén, and fast magnetosonic waves.

HMHD spectrum is homotopic to that of ideal MHD, with no extra wave branches.

The wave structure exhibits nontrivial topology characterized by Weyl points and nonzero Chern numbers.

Abstract

Hall Magnetohydrodynamics (HMHD) extends ideal MHD by incorporating the Hall effect via the induction equation, making it more accurate for describing plasma behavior at length scales below the ion skin depth. Despite its importance, a comprehensive description of the eigenmodes in HMHD has been lacking. In this work, we derive the complete spectrum and eigenvectors of HMHD waves and identify their underlying topological structure. We prove that the HMHD wave spectrum is homotopic to that of ideal MHD, consisting of three distinct branches: the slow magnetosonic-Hall waves, the shear Alfv\'en-Hall waves, and the fast magnetosonic-Hall waves, which continuously reduce to their ideal MHD counterparts in the limit of vanishing Hall parameter. Contrary to a recent claim, we find that HMHD does not admit any additional wave branches beyond those in ideal MHD. The key qualitative difference lies in the topological nature of the HMHD wave structure: it exhibits nontrivial topology characterized by a Weyl point-an isolated eigenmode degeneracy point-and associated nonzero Chern numbers of the eigenmode bundles over a 2-sphere in k-space surrounding the Weyl point.