Infinite mass boundary conditions for Dirac operators

Abstract

We study a self-adjoint realization of a massless Dirac operator on a bounded connected domain Ω ⊂ R 2 which is frequently used to model graphene. In particular, we show that this operator is the limit, as M → ∞, of a Dirac operator defined on the whole plane, with a mass term of size M supported outside Ω.Equation (1) gives rise to a whole family of different boundary conditions (see Equation (2) below). In this present work we focus on one of these self-adjoint realizations, denoted by H ∞ , which corresponds to the so-called infinite mass boundary conditions. In the physics literature, the operator H ∞ has gained renewed interest due to its application to model quantum dots in graphene [6,7,15,11,14,12,2].Let H M be the Dirac operator defined on R 2 with a mass M on R 2 \ Ω, and 0 inside Ω. In [4] it was shown that certain plane-wave solutions of the eigenvalue equation H M ψ = Eψ, in the limit M → ∞, satisfy the same boundary conditions as the eigenfunctions of H ∞ . The main result of this work, Theorem 1, is the convergence, in the sense of spectral projections, of H M towards H ∞ .1.1. Definitions and main result. Let us introduce some notation used throughout this article. We denote by Ω ⊂ R 2 a bounded connected domain with boundary ∂Ω ∈ C 3 of length L > 0. We parametrize ∂Ω by the curve γ : [0, L] → ∂Ω in its arc-length, i.e., |γ ′ (s)| = 1. For a given self-adjoint operator H, we denote by σ(H) its spectrum, and by E I (H) its spectral projection on the set I ⊂ R. We use the symbols ·, · and (·, ·) to denote the scalar products in L 2 and C 2 , respectively.

Dirac Operators on Hypersurfaces as Large Mass Limits

“…For the lower bound we first decouple the two sides of Ω in order to deal separately with Ω and its exterior, then it is easily seen that the exterior does not contribute to the lowest eigenvalues, while the part in Ω appears to be monotonically increasing in M and then easily handled with the help of the monotone convergence. The overall scheme here is very close to the one used in [19] for the two-dimensional case.…”

Resolvent Convergence to Dirac Operators on Planar Domains

“…This result is stated in [22,Lemma 4] for a bounded and regular domain Ω. However, its proof extends easily to unbounded domains.…”

“…where in the last step we used the hypothesis on m. We find the bounds (21) and (22) after inserting ϕ = R Proposition 3.4 (A priori lower bound [22]). There exist constants c < ∞ and m ′ 0 > 1 such that for all ϕ ∈ H 1 (R 2 , C 2 )…”

“…The strategy used in [22] was based on the mini-max principle, which is only appropriate for bounded domains. In this work we extend the results from [22] in several directions.…”

A Sharp Upper Bound on the Spectral Gap for Graphene Quantum Dots

“…Note that infinite mass boundary conditions for the Dirac operator arise when one considers the Dirac operator on the whole Euclidean plane R 2 with an "infinite mass" outside a bounded domain and zero mass inside it. This is mathematically justified in [5,34] (see also [4] for a three-dimensional version and [24] for a generalization to any dimension). For this reason, these boundary conditions can be viewed as the relativistic counterpart of Dirichlet boundary conditions for the Laplacian.…”