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Topology of 2D Dirac operators with variable mass and an application to shallow-water waves

Sylvain Rossi, Alessandro Tarantola

Abstract

A Dirac operator on the plane with constant (positive) mass is a Chern insulator, sitting in class D of the Kitaev table. Despite its simplicity, this system is topologically ill-behaved: the non-compact Brillouin zone prevents definition of a bulk invariant, and naively placing the model on a manifold with boundary results in violations of the bulk-edge correspondence (BEC). We overcome both issues by letting the mass spatially vary in the vertical direction, interpolating between the original model and its negative-mass counterpart. Proper bulk and edge indices can now be defined. They are shown to coincide, thereby embodying BEC. The shallow-water model exhibits the same illnesses as the 2D massive Dirac. Identical problems suggest identical solutions, and indeed extending the approach above to this setting yields proper indices and another instance of BEC.

Topology of 2D Dirac operators with variable mass and an application to shallow-water waves

Abstract

A Dirac operator on the plane with constant (positive) mass is a Chern insulator, sitting in class D of the Kitaev table. Despite its simplicity, this system is topologically ill-behaved: the non-compact Brillouin zone prevents definition of a bulk invariant, and naively placing the model on a manifold with boundary results in violations of the bulk-edge correspondence (BEC). We overcome both issues by letting the mass spatially vary in the vertical direction, interpolating between the original model and its negative-mass counterpart. Proper bulk and edge indices can now be defined. They are shown to coincide, thereby embodying BEC. The shallow-water model exhibits the same illnesses as the 2D massive Dirac. Identical problems suggest identical solutions, and indeed extending the approach above to this setting yields proper indices and another instance of BEC.
Paper Structure (11 sections, 7 theorems, 122 equations, 4 figures, 1 table)

This paper contains 11 sections, 7 theorems, 122 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Setup: Two-dimensional Dirac Hamiltonians with (non)-constant mass
  3. Bulk index
  4. Edge index and bulk-edge correspondence
  5. Application: Rotating shallow-water model
  6. Conclusions and future directions
  7. Acknowledgements
  8. Proof of Proposition \ref{['prop:EdgeIndex']}
  9. Essential spectrum of the shallow-water Hamiltonian
  10. Interface states of the shallow-water model, complete calculation
  11. Failure of BEC: Dirac model on the half-plane

Key Result

Proposition 1

The quantity $C \hbar (\mathcal{E}_+) - C \hbar (\mathcal{E}_-)$ is a genuine topological invariant, namely there exists a bundle $\mathcal{E}$ over compact base space $\tilde{\mathbb{D}} \simeq S^2$ such that

Figures (4)

  • Figure 1: From left to right: position space before folding; position space after folding and Alice standing close to the equator; Alice after moving towards the "upper bulk", now looking down at the lower rim.
  • Figure 2: Visualization of the compactification procedure. Left panel: oppositely oriented planar Brillouin zones $\mathbb{E}^*_\pm$. Center panel: disks $\mathbb{D}_\pm$. Right panel: $\mathbb{D}_\pm$ joined along the boundary $\partial \mathbb{D}_+ = \partial \mathbb{D}_-$ to form an oriented sphere $\tilde{\mathbb{D}} \cong S^2$.
  • Figure 3: Left panel: Illustration of intersection numbers and related conventions. In this example, $I (\mu_\epsilon,\omega_1) = +1$ and $I (\mu_\epsilon,\omega_2) = -1$. By Eq. \ref{['eq:EdgeIndex']}, $\mathcal{I}^\# = -I(\mu_\epsilon, \omega_1) -I (\mu_\epsilon, \omega_2) = -1 +1 = 0$. Middle panel: contribution of $\omega_L (k_1)$ to the edge index. Right panel: allowed region ("light cone") for bound states. If there exist bound states besides the one with dispersion relation $\omega_L$, their net contribution to $\mathcal{I}^\#$ is zero.
  • Figure 4: Edge spectrum and edge index when $f(x_2) = f \operatorname{sgn} (x_2)$. From left to right: Edge channels; Edge index as per Def. \ref{['def:EdgeIndex']}; Edge index in a rescaled picture $\omega / \sqrt{k_1^2 + f^2}$ against $\tanh (k_1)$.

Theorems & Definitions (25)

  • Remark 1
  • Definition 1
  • Proposition 1
  • Remark 2
  • Remark 3
  • Definition 2
  • Remark 4
  • Proposition 2
  • Remark 5
  • Remark 6
  • ...and 15 more