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The dot DP∨,1,5σe0,12ω1∨ of C2. The light blue triangle represents the fundamental domain Fσe(0). The boundary black dashed lines represent the Dirichlet walls Hσe(0). The point set FP∨,5σe0,12ω1∨, representing possible positions of the quantum particle, is formed by the dark dots. The lines connecting the dots symbolise the nearest neighbour coupling characterised by the hopping operator A^ω2∨,5σe0,12ω1∨.
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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...
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Citations
... Tight-binding models of electron propagation represent fundamental tools for description of electronic properties of modern materials [6,22]. Besides utilizing the standard periodic boundary conditions, inclusions of the Dirichlet/Neumann boundary behaviours through dual-root and dual-weight discrete Fourier-Weyl transforms have been recently achieved [4,5]. The current generalized versions of A 1 × A 1 even dual-weight Fourier-Weyl transforms conceivably lead to explicitly solvable Hamiltonian descriptions of the quantum particle propagation on the underlying point sets while subjected to non-standard, transitional boundary conditions. ...
The even subgroup of the Weyl group associated with the crystallographic root system A1×A1 induces two-variable even Weyl orbit functions which form the kernels of the developed discrete Fourier–Weyl transforms. The finite point and label sets of the discrete trigonometric transforms are formed by rectangular fragments of the A1×A1 admissibly shifted weight lattices in the Euclidean plane. Sixteen types of the point and label sets are listed and the related discrete orthogonality relations of the even Weyl orbit functions are demonstrated. The forward and backward transforms together with the linked interpolation formulas and orthogonal transform matrices are presented and exemplified.
... Periodic boundary conditions serve as ubiquitously utilized assumptions which provide finite labellings of the electronic stationary states [39,43]. From a viewpoint of crystallographic root systems and associated affine Weyl groups, combining presupposed rescaled dual root lattice periodicity together with Weyl group (anti)symmetry produces descriptions of a quantum particle propagating on rescaled Weyl group invariant lattices inside rescaled closures of the Weyl alcoves and subjected to unique mixtures of Dirichlet and Neumann boundary conditions [4,5,19]. Replacing Weyl group symmetries with their even subgroup versions [18,29], boundary conditions of the currently considered quantum particle propagations on dual weight lattices constitute transitions between the dual-root periodic boundary conditions and Dirichlet/Neumann conditions. The even dual-weight Fourier-Weyl transforms of crystallographic root systems corresponding to the affine even Weyl groups [17,18] form the mathematical apparatus that provides an efficient description of the presented even dual-weight quantum dots. ...
... Each non-zero value of the even dual-weight hopping function inside the fundamental domain of the even Weyl group induces the associated even hopping operator which approximates the particle's jumps to lattice neighbours of the corresponding degree as well as incorporates the boundary conditions into the resulting Hamiltonian. While the amplitude reflections from the boundary walls within the non-subgroup models are characterized via the sign χ -functions and coupling sets [4,5,19], the even sign χ -functions and even coupling sets also consistently encompass the generalized periodic as well as mixed boundary conditions. The energy spectra of the ensuing Hamiltonians of the tight-binding type are explicitly expressed through sums of products of the even dualweight hopping functions and even Weyl orbit functions. ...
... Similarly to the non-subgroup discrete Fourier-Weyl transforms and their one-dimensional archetypes [37], the discrete dual-weight orthogonality relations of the even Weyl orbit functions are weighted and normalized by the even dual-weight and weight counting functions [17,18]. Analogously to the dual-root and dual-weight propagation models in the rescaled Weyl alcoves [4,5], it appears that besides abstract relevance of the counting functions, measurable consequences affecting the particle's behaviour on the Neumann type boundaries of the rescaled even Weyl alcoves are inferred. ...
Even subgroups of affine Weyl groups corresponding to irreducible crystallographic root systems characterize families of single-particle quantum systems. Induced by primary and secondary sign homomorphisms of the Weyl groups, free propagations of the quantum particle on the refined dual weight lattices inside the rescaled even Weyl alcoves are determined by Hamiltonians of tight-binding types. Described by even hopping functions, amplitudes of the particle’s jumps to the lattice neighbours are together with diverse boundary conditions incorporated through even hopping operators into the resulting even dual-weight Hamiltonians. Expressing the eigenenergies via weighted sums of the even Weyl orbit functions, the associated time-independent Schrödinger equations are exactly solved by applying the discrete even Fourier–Weyl transforms. Matrices of the even Hamiltonians together with specifications of the complementary boundary conditions are detailed for the C2 and G2 even dual-weight models.
... 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. ...
Tight-binding models of electron propagation in single-layer triangular graphene quantum dots with armchair and zigzag edges are developed. The electron hoppings to the nearest and next-to-nearest neighbours on the honeycomb lattice as well as interactions with the confining Dirichlet and Neumann walls are incorporated into the resulting tight-binding Hamiltonians. Associated to the irreducible crystallographic root system A 2 , the armchair and zigzag honeycomb Weyl orbit functions together with the related discrete Fourier–Weyl transforms provide explicit exact forms of the electron wave functions and energy spectra. The electronic probability densities corresponding to the armchair and zigzag dots are evaluated and their contrasting behaviour exemplified.
