Wave Topology in Hall MHD

TL;DR

The paper formulates Hall MHD as a Hermitian Hamiltonian system and derives the full HMHD wave spectrum via Cardano's method, revealing three Hall-modified branches that continuously deform to the ideal MHD branches as the Hall parameter $ au$ vanishes. It shows a Weyl point in the HMHD spectrum and proves nontrivial topology of the corresponding eigenbundles, quantified by Chern numbers $C=\pm1$, while demonstrating that no additional wave branches arise beyond those of ideal MHD. A key result is the homotopy between HMHD and ideal MHD spectra: the branches map one-to-one under $ au\to0$, resolving prior claims of extra modes. The work suggests that topological features in homogeneous HMHD extend to inhomogeneous plasmas, potentially producing topologically protected edge modes such as the topological Alfvén sound wave in parity-time symmetric Hall MHD systems.

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 Alfvén-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.

Figures (3)

• Figure 1: 2D line plots of the eigenfrequencies $\omega_{0}$ (red), $\omega_{1}$ (blue), and $\omega_{2}$ (green) from Eq. (\ref{['Eq:eigenfrequencies']}) as functions of $k_{z}$ for various $k_{\perp}$ values. The Weyl point is shown for both cases of $\zeta > 1\,(\color{blue}+\color{black})$ and $\zeta < 1\, (\color{blue}-\color{black})$. The Weyl point occurs at $(k_{\perp},k_{z}) = (0,k_{z0}^{\color{blue}\pm\color{black}})$, where $k_{z0}^{\color{blue}\pm\color{black}}\equiv \color{blue}\pm\color{black}(\zeta\!-\!\zeta^{-1})/\tau$ and $\omega_{\ast}^{\color{blue}\pm\color{black}}\equiv \zeta k_{z0}^{\color{blue}\pm\color{black}}$.
• Figure 2: 3D surface plots of the eigenfrequencies $\omega_{0}$ (red), $\omega_{1}$ (blue), and $\omega_{2}$ (green) from Eq. (\ref{['Eq:eigenfrequencies']}) as functions of $(k_{\perp},k_{z})$. The Weyl point is shown for both cases of $\zeta > 1\,(\color{blue}+\color{black})$ and $\zeta < 1\, (\color{blue}-\color{black})$. The Weyl point occurs at $(k_{\perp},k_{z}) = (0,k_{z0}^{\color{blue}\pm\color{black}})$, where $k_{z0}^{\color{blue}\pm\color{black}}\equiv \color{blue}\pm\color{black}(\zeta\!-\!\zeta^{-1})/\tau$ and $\omega_{\ast}^{\color{blue}\pm\color{black}}\equiv \zeta k_{z0}^{\color{blue}\pm\color{black}}$.
• Figure 3: Dirac cones at the Weyl point in $(k_{x},k_{y})$ space between $\omega_{2}$ (green) and $\omega_{1}$ (blue) for $\zeta > 1\,(\color{blue}+\color{black})$, and between $\omega_{1}$ (blue) and $\omega_{0}$ (red) for $\zeta < 1\,(\color{blue}-\color{black})$, corresponding to $(k_{x},k_{y},k_{z}) = (0,0,k_{z0}^{\color{blue}\pm\color{black}})$.