2012/03/31 by Merab Eliashvili, M Eliashvili, George I. Japaridze +3 · 2 citations
Mathematics · Physics and Astronomy · #Algebraic structures and combinatorial models #Eigenfunction #Eigenvalues and eigenvectors #Hamiltonian (control theory) #Lattice (music) #Quantum #Quantum Mechanics and Non-Hermitian Physics #Quantum system #Symmetry group #Topological Materials and Phenomena #cond-mat.mes-hall #math-ph #math.MP
paper · pdf · doi:10.1088/1751-8113/45/39/395305
published in Journal of Physics A Mathematical and Theoretical 45(39), 395305 (Institute of Physics) · 11 pages, 4 figures
arxiv created 2012/09/03 · openalex publication_date 2012/09/12 · arxiv updated 2012/09/13 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
The tight-binding model of quantum particles on a honeycomb lattice is investigated in the presence of homogeneous magnetic field. Provided the magnetic flux per unit hexagon is rational of the elementary flux, the one-particle Hamiltonian is expressed in terms of the generators of the quantum group Uq(sl2). Employing the functional representation of the quantum group Uq(sl2) the Harper equation is rewritten as a systems of two coupled functional equations in the complex plane. For the special values of quasi-momentum the entangled system admits solutions in terms of polynomials. The system is shown to exhibit certain symmetry allowing to resolve the entanglement, and basic single equation determining the eigenvalues and eigenstates (polynomials) is obtained. Equations specifying locations of the roots of polynomials in the complex plane are found. Employing numerical analysis the roots of polynomials corresponding to different eigenstates are solved out and the diagrams exhibiting the ordered structure of one-particle eigenstates are depicted.