197 research outputs found
The quantum group, Harper equation and the structure of Bloch eigenstates on a honeycomb lattice
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 . Employing the functional representation of the quantum group
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.Comment: 11 pages, 4 figure
Synthetic Gauge Fields for Vibrational Excitations of Trapped ions
The vibrations of a collection of ions in a microtrap array can be described
in terms of hopping phonons. We show theoretically that the vibrational
couplings may be tailored by using a gradient of the microtrap frequencies,
together with a periodic driving of the trapping potential. These ingredients
allow us to induce effective gauge fields on the vibrational excitations, such
that phonons mimic the behavior of charged particles in a magnetic field. In
particular, microtrap arrays are ideally suited to realize the famous
Aharonov-Bohm effect, and observe the paradigmatic edge states typical from
quantum-Hall samples and topological insulators.Comment: replaced with published versio
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