Quantum Particle on Dual Weight Lattice in Weyl Alcove

Abstract

Families of discrete quantum models that describe a free non-relativistic quantum particle propagating on rescaled and shifted dual weight lattices inside closures of Weyl alcoves are developed. The boundary conditions of the presented discrete quantum billiards are enforced by precisely positioned Dirichlet and Neumann walls on the borders of the Weyl alcoves. The amplitudes of the particle’s propagation to neighbouring positions are determined by a complex-valued dual-weight hopping function of finite support. The discrete dual-weight Hamiltonians are obtained as the sum of specifically constructed dual-weight hopping operators. By utilising the generalised dual-weight Fourier–Weyl transforms, the solutions of the time-independent Schrödinger equation together with the eigenenergies of the quantum systems are exactly resolved. The matrix Hamiltonians, stationary states and eigenenergies of the discrete models are exemplified for the rank two cases C2 and G2.

On electron propagation in triangular graphene quantum dots

“…with the states associated to the armchair vectors |(λ 2 , λ 1 ); A, σ, ± for which λ 1 = λ 2 and (λ 2 , λ 1 ) ∈ L σ,± A,M . This type of energy degeneracy is similarly observed within the current description of the zigzag momentum bases (134) as well as for the root and weight A 2 lattice models with real-valued hopping functions [4,5]. Possible lifting of this degeneracy by incorporating complex-valued factors in the tight-binding Hamiltonians (110) and (132), which appear for instance in presence of an external magnetic field [14,21], merits further investigation.…”

“…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].…”

“…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]. 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. 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.…”

“…The developed exactly solvable models of the electron propagation in both zigzag and armchair triangular graphene quantum dots surrounded by ideal Dirichlet and Neumann walls offer significant theoretical insight into behaviour of a quantum particle on pseudolattice fragments as well as foundation for further investigations of the graphene and other honeycomb structured [1,7] quantum dots. Even though theoretical research and experimental realization of the Neumann boundary conditions pertaining to the triangular graphene dots are currently less prominent [49], simultaneous mathematical exposition, along with parallel presentation of both Neumann and Dirichlet cases, invokes direct analogies with the A 2 lattice models [4,5] for the armchair dots and provides essential context for the uniquely contrasting characteristics of the zigzag dots [2,52,62]. Most remarkably, the calculated electronic probability densities of the Neumann boundary zigzag conduction bands coincide with the densities of the Dirichlet boundary zigzag valence bands and vice versa.…”