Discrete cosine and sine transforms generalized to honeycomb lattice II. Zigzag boundaries

Abstract

The discrete cosine and sine transforms are generalized to a triangular fragment of the honeycomb lattice with zigzag boundaries. The zigzag honeycomb point sets are constructed by subtracting the weight lattice from the refined root lattice points of the crystallographic root system A2. The two-variable (anti)symmetric orbit functions of the Weyl group of A2, discretized simultaneously on the triangular fragments of the root and weight lattices, induce a novel parametric family of zigzag extended Weyl and Hartley orbit functions. As specific linear combinations of the original orbit functions, the zigzag extended orbit functions retain the Neumann and Dirichlet boundary conditions. Three types of discrete complex Fourier–Weyl transforms and real-valued Hartley–Weyl transforms are detailed. The corresponding unitary transform matrices and interpolating behavior of the discrete transforms are exemplified.

On electron propagation in triangular graphene quantum dots

“…For an arbitrary magnifying factor M ∈ N, the honeycomb lattice Z M is introduced subtractively [28] as follows,…”

“…The purpose of this article is to formulate and analytically solve classes of tight-binding models [8,21] that describe an electron propagating in single-layer triangular graphene quantum dots with armchair and zigzag edges [2,16,24,61]. Extending the results for the armchair dots [53] via the honeycomb Weyl orbit functions and the corresponding discrete Fourier-Weyl transforms [26,28], explicit exact forms of the electron wave functions and energy spectra are constructed.…”

“…The point sets describing the atomic positions of the armchair and zigzag triangular graphene dots are advantageously expressed via combinations of A 2 rescaled root and weight lattices [26,28]. Linked to the affine Weyl groups [3,32], discrete Fourier-Weyl transforms utilizing Weyl orbit functions [36,37] have been recently developed on general finite fragments of root and weight lattices [27,30,31,40] as well as subsequently adapted to triangular honeycomb dots [26,28].…”

“…The point sets describing the atomic positions of the armchair and zigzag triangular graphene dots are advantageously expressed via combinations of A 2 rescaled root and weight lattices [26,28]. Linked to the affine Weyl groups [3,32], discrete Fourier-Weyl transforms utilizing Weyl orbit functions [36,37] have been recently developed on general finite fragments of root and weight lattices [27,30,31,40] as well as subsequently adapted to triangular honeycomb dots [26,28]. The tight-binding models of quantum particle propagating on the (dual) root and weight lattices in Weyl alcoves use concepts of coupling sets, εand χ-functions to incorporate the interactions of the particle with the boundary walls into the Hamiltonians and are exactly solvable for any fixed degree of coupling by exploiting the product-to-sum formulas of the Weyl orbit functions [4,5].…”

“…Nevertheless, the description up to the second-nearestneighbour coupling already incorporates the intrinsic effects observed for the lattice models [4,5] and necessitates the forthcoming comprehensive summary of the pertinent mathematical background specialized to the root system A 2 . Since the honeycomb Weyl orbit functions along with the associated honeycomb Fourier-Weyl transforms have already been constructed [26,28], only the honeycomb product-to-sum decomposition formulas remain to be established.…”