[Paper Review] Continuum Schroedinger operators for sharply terminated graphene-like structures

Charles Fefferman, Michael I. Weinstein|arXiv (Cornell University)|Oct 8, 2018

Topological Materials and Phenomena50 references4 citations

TL;DR

This paper establishes a rigorous mathematical connection between continuum Schrödinger operators and tight-binding models for sharply terminated graphene-like structures in the strong binding regime. It proves scaled resolvent convergence to a discrete tight-binding Hamiltonian and rigorously constructs edge states localized at zigzag edges, demonstrating their existence via a flat-band mechanism in the continuum limit.

ABSTRACT

We study the single electron model of a semi-infinite graphene sheet interfaced with the vacuum and terminated along a zigzag edge. The model is a Schroedinger operator acting on $L^2(\mathbb{R}^2)$: $H^λ_{ m edge}=-Δ+λ^2 V_\sharp$, with a potential $V_\sharp$ given by a sum of translates an atomic potential well, $V_0$, of depth $λ^2$, centered on a subset of the vertices of a discrete honeycomb structure with a zigzag edge. We give a complete analysis of the low-lying energy spectrum of $H^λ_{ m edge}$ in the strong binding regime ($λ$ large). In particular, we prove scaled resolvent convergence of $H^λ_{ m edge}$ acting on $L^2(\mathbb{R}^2)$, to the (appropriately conjugated) resolvent of a limiting discrete tight-binding Hamiltonian acting in $l^2(\mathbb{N}_0;\mathbb{C}^2)$. We also prove the existence of {\it edge states}: solutions of the eigenvalue problem for $H^λ_{ m edge}$ which are localized transverse to the edge and pseudo-periodic (propagating or plane-wave like) parallel to the edge. These edge states arise from a "flat-band" of eigenstates the tight-binding Hamiltonian.

Motivation & Objective

• To rigorously derive the emergence of edge states in semi-infinite graphene-like structures with a sharp zigzag termination.
• To establish the convergence of the continuum Schrödinger operator's resolvent to a discrete tight-binding Hamiltonian in the strong binding regime.
• To analyze the low-lying energy spectrum, including both discrete edge states and continuous spectrum, in the context of a sharply terminated honeycomb lattice.
• To provide a mathematical foundation for topological edge states in graphene-based systems, particularly those protected by symmetry and nontrivial Berry phases.
• To bridge the gap between continuum quantum mechanics and discrete tight-binding models in the context of edge state formation at zigzag boundaries.

Proposed method

• Formulates a Schrödinger operator $ H^\lambda_{\rm edge} = -\Delta + \lambda^2 V_{\sharp} $ on $ L^2(\mathbb{R}^2) $, where $ V_{\sharp} $ is a sum of translated atomic potential wells on a honeycomb lattice with a zigzag edge.
• Applies the strong binding regime ($ \lambda \to \infty $) to derive effective dynamics via asymptotic analysis of the resolvent.
• Uses a conjugated resolvent framework to relate the continuum operator to a discrete tight-binding Hamiltonian on $ \ell^2(\mathbb{N}_0; \mathbb{C}^2) $.
• Employs exponential decay estimates for Wannier functions and potential matrix elements to control non-local couplings.
• Analyzes the spectral structure using Floquet-Bloch theory and identifies edge states through the flat-band mechanism of the tight-binding model.
• Applies symmetry and localization arguments to prove the existence of edge states that are pseudo-periodic along the edge and localized transverse to it.

Experimental results

Research questions

• RQ1 How does the continuum Schrödinger operator for a sharply terminated graphene-like structure converge to a discrete tight-binding model in the strong binding regime?
• RQ2 What conditions lead to the existence of edge states in the spectrum of the continuum Schrödinger operator at a zigzag edge?
• RQ3 How is the spectral structure of the continuum model related to the flat-band eigenstates of the tight-binding Hamiltonian?
• RQ4 What role does the Berry-Zak phase play in determining the existence of edge states at different edge orientations?
• RQ5 Can edge states be rigorously constructed in the continuum limit, and how do they behave in terms of localization and propagation?

Key findings

• The resolvent of the continuum Schrödinger operator $ H^\lambda_{\rm edge} $ converges, in a scaled sense, to the resolvent of a discrete tight-binding Hamiltonian acting on $ \ell^2(\mathbb{N}_0; \mathbb{C}^2) $ as $ \lambda \to \infty $.
• Edge states—localized transverse to the zigzag edge and pseudo-periodic along it—exist in the spectrum of $ H^\lambda_{\rm edge} $ for a subinterval of quasi-momenta $ k_\parallel \in [0, 2\pi) $.
• These edge states arise from a flat band in the tight-binding model, and their existence is robust under the strong binding limit.
• The matrix elements $ \mathscr{I}(\sigma, \mathbf{r}, \tilde{\sigma}, \tilde{\mathbf{r}}) $ decay exponentially with $ \lambda $, with bounds depending on the distance from the edge and the band structure.
• The analysis confirms that edge states do not exist at armchair terminations, consistent with the non-vanishing Berry-Zak phase only for zigzag edges.
• The paper establishes a rigorous mathematical link between the continuum Schrödinger equation and the tight-binding approximation in the strong binding limit, validating the latter for edge state physics.