The quantum group, Harper equation and the structure of Bloch eigenstates on a honeycomb lattice

TL;DR

This paper explores the structure of Bloch eigenstates in a honeycomb lattice under a magnetic field using quantum groups, deriving polynomial solutions and analyzing their roots to understand eigenstate organization.

Contribution

It introduces a novel approach by expressing the Hamiltonian with quantum group generators and deriving polynomial solutions for eigenstates in a honeycomb lattice system.

Findings

Polynomial solutions for eigenstates are obtained at special quasi-momenta.

Root locations of polynomials reveal the ordered structure of eigenstates.

Symmetries allow reduction to a single eigenvalue equation.

Abstract

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 $U_q(sl_2)$. Employing the functional representation of the quantum group $U_q(sl_2)$ 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.