Graphene Dots via Discretizations of Weyl-Orbit Functions

Quantum Particle on Dual Weight Lattice in Weyl Alcove

“…The quantum billiards on 2D triangles [10][11][12] and 3D Weyl chamber [13] realise the closest versions of the current simplex-shaped models. From the viewpoint of position restrictions, the discrete quantum billiards [4,14] involve the propagating particle on quantum dots [15][16][17]. Single particle properties of the quantum dots are commonly studied via the discrete tight-binding Hamiltonians [3,15,17].…”

On electron propagation in triangular graphene quantum dots

“…Proof. The explicit forms of the discretized zigzag honeycomb orbit functions (102), zigzag sign function (29) and normalization function (96) are used to directly verify for the interior weights λ ∈ L σ e M the following equality,…”

“…In contrast, the electron stationary states and energy spectra of the triangular graphene dots with zigzag edges remain accessible mostly by numerical computations [2,24,61]. It appears that for a uniform characterization of the exact electron wave functions and energy spectra of both armchair and zigzag triangular graphene dots, interactions of the electron with ideally positioned Dirichlet or Neumann boundary walls need to be specifically embedded into the tight-binding Hamiltonians [4,5,29]. In order to achieve such rigorous Hamiltonian descriptions as well as thoroughly utilize underlying symmetries for finding their exact solutions, both triangular armchair and zigzag graphene dots are studied in the context of the affine Weyl group associated to the irreducible crystallographic root system A 2 [3,32].…”

“…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]. The Hamiltonian formulation for the armchair graphene dots with the nearest neighbour coupling employing the Weyl orbit functions has already been partially undertaken [29]. In order to partly facilitate the current mathematical exposition, the presented models on the honeycomb (pseudo)lattice assume the nearest and next-to-nearest electron hopping only.…”

Connecting (Anti)Symmetric Trigonometric Transforms to Dual-Root Lattice Fourier–Weyl Transforms

“…Moreover, the cubature rules [23] of the multivariate Chebyshev polynomials, that are obtained from the (anti)symmetric trigonometric functions [9,10] and associated with the Jacobi polynomials [5,24,25], are further intertwined with the Lie theoretical approach [8,26,27]. The role of the 2D and 3D Fourier-Weyl transforms as tools for solutions of the lattice vibration and electron propagation models [28,29] implies comparable function and direct applicability of the (anti)symmetric trigonometric transforms in solid state physics [30,31] and quantum field theory [32]. The potential diverse applications of both types of discrete transforms also involve image compression [33], laser optics [34], fluid flows [35], magnetostatic modeling [36], and micromagnetic simulations [37].…”