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Edge states for tight-binding operators with soft walls

Camilo Gómez Araya, David Gontier, Hanne Van Den Bosch

TL;DR

The paper develops a rigorous, geometry-agnostic framework for edge states in tight-binding models terminated by soft walls. By proving a spectral flow as the wall shifts and showing it equals the negative count of bulk Bloch bands below a given energy, it establishes a universal link between wall position and edge modes across 1D and 2D TB systems. The approach hinges on reducing general periodic Hamiltonians to Jacobi-type models, using dislocations and cut limits to compute the flow explicitly, and then lifting the results to higher dimensions via Fourier decomposition along the wall. The findings imply robust, wall-position-dependent edge spectra in a broad class of materials, with concrete demonstrations in the SSH chain and graphene Wallace model, and provide quantitative lower bounds on edge-state numbers in bulk gaps. This has potential implications for designing materials and devices with tunable edge transport by soft confinement and cut orientation.

Abstract

We study one- and two-dimensional periodic tight-binding models under the presence of a potential that grows to infinity in one direction, hence preventing the particles to escape in this direction (the soft wall). We prove that a spectral flow appears in these corresponding edge models, as the wall is shifted. We identity this flow as a number of Bloch bands, and provide a lower bound for the number of edge states appearing in such models.

Edge states for tight-binding operators with soft walls

TL;DR

The paper develops a rigorous, geometry-agnostic framework for edge states in tight-binding models terminated by soft walls. By proving a spectral flow as the wall shifts and showing it equals the negative count of bulk Bloch bands below a given energy, it establishes a universal link between wall position and edge modes across 1D and 2D TB systems. The approach hinges on reducing general periodic Hamiltonians to Jacobi-type models, using dislocations and cut limits to compute the flow explicitly, and then lifting the results to higher dimensions via Fourier decomposition along the wall. The findings imply robust, wall-position-dependent edge spectra in a broad class of materials, with concrete demonstrations in the SSH chain and graphene Wallace model, and provide quantitative lower bounds on edge-state numbers in bulk gaps. This has potential implications for designing materials and devices with tunable edge transport by soft confinement and cut orientation.

Abstract

We study one- and two-dimensional periodic tight-binding models under the presence of a potential that grows to infinity in one direction, hence preventing the particles to escape in this direction (the soft wall). We prove that a spectral flow appears in these corresponding edge models, as the wall is shifted. We identity this flow as a number of Bloch bands, and provide a lower bound for the number of edge states appearing in such models.
Paper Structure (35 sections, 30 theorems, 187 equations, 9 figures)

This paper contains 35 sections, 30 theorems, 187 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Spectral flows in one-dimensional models
  3. Generalities for bulk and edge Hamiltonians
  4. Bulk periodic Hamiltonians
  5. Soft wall operators
  6. Computation of the spectral flow: reduction to the Jacobi case
  7. The spectrum at fixed $t_0$, proof of Theorem \ref{['th:main_fix_t0']}
  8. Example : the SSH model
  9. Presentation of the model, and basic facts
  10. Numerical simulations for the SSH chain
  11. Computation of spectral flows in the Jacobi case
  12. Reduction to the cut case
  13. Dislocated Jacobi operators
  14. Computation of the spectral flow for the dislocated model
  15. The finite periodic dislocated model
  16. ...and 20 more sections

Key Result

Theorem 1.1

Let $w : \mathbb{R} \mapsto {\mathcal{S}}_N$ be a Lipschitz function satisfying eq:wall_def_intro. For all $E \in \mathbb{R} \setminus \sigma_{\rm{bulk}}$, the energy $E$ is in the essential gaps of all operators $H^\sharp(t)$, and where we recall that ${\mathcal{N}}(E)$ is the number of Bloch bands below $E$ for the bulk--operator.

Figures (9)

  • Figure 1: Schematic illustration of the models in the proof. In the first figure, as $t$ varies from $0$ to $1$, the wall is pushed one cell to the right. We prove that all spectral flows coincide for these four scenarios, and explicitly compute the third one.
  • Figure 2: Left: the function $w(x) := \frac{1}{2} \left(\sqrt{x^2 + 1} - x \right)$, with $N=1$. Right: the spectrum of the corresponding operator $W(t)$, as a function of $t$ (the $n=0$ case is in red). As $t$ increases, the wall moves to the right. Here, $t \mapsto W(t)$ is operator increasing.
  • Figure 3: Left: illustration of the SSH model. Right: its band structure for different values of $\delta := | |J_1| - |J_2| |$.
  • Figure 4: Numerics for the spectrum of $t \mapsto H^\sharp(t)$ in the SSH model with the potential $w_1(x)$ , for different values of $\nu$. From left to right, $\nu = 0.5$, $\nu = 1$, $\nu = 5$ and $\nu = 10$. The essential spectrum is displayed in grey, and the edge spectrum in red. For $E$ in the gap between the $2$ bands, ${\mathcal{N}}(E) = 1$ and the spectral flow is $-1$, above the bands, ${\mathcal{N}}(E) = 2$ and the spectral flow equals $-2$. When the Lipschitz constant increases, the curves become steeper.
  • Figure 5: Our convention for the zigzag direction (left), armchair direction with $(n,m) = (-1, 1)$ (middle), and another angle with $(n,m) = (2, -1)$ (right). The shaded cell represents $\Gamma$ (resp. $\widetilde{\Gamma}$, $\overline{\Gamma}$).
  • ...and 4 more figures

Theorems & Definitions (54)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Lemma 2.1
  • proof
  • Remark 2.2
  • Definition 2.3
  • Proposition 2.4
  • proof : Proof of Proposition \ref{['prop:basics_intro']}
  • ...and 44 more