Quantum particle on dual root lattice in Weyl alcove

Abstract

Classes of discrete quantum models that describe a free non-relativistic quantum particle propagating on rescaled and shifted dual root lattices inside closures of Weyl alcoves are constructed. Boundary conditions of the discrete quantum billiard systems on the borders of the Weyl alcoves are controlled by specific combinations of Dirichlet and Neumann walls that result from sign homomorphisms and admissible shifts inherent in generalized dual root lattice Fourier–Weyl transforms. The amplitudes of the particle’s jumps to neighbouring positions are controlled by a complex-valued dual root lattice hopping function with finite support. The solutions of the time-independent Schrödinger equation together with the eigenenergies of the quantum systems are explicitly determined. The matrix Hamiltonians and eigenenergies of the discrete models are exemplified for the rank two cases A 2 and C 2.

On electron propagation in triangular graphene quantum dots

“…Calculated from exact energy relation (115), the eigenenergies 4 of the electron in the armchair graphene dot G 1 A,4 are determined as…”

“…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.…”