851 research outputs found
Honeycomb Lattice Potentials and Dirac Points
We prove that the two-dimensional Schroedinger operator with a potential
having the symmetry of a honeycomb structure has dispersion surfaces with
conical singularities (Dirac points) at the vertices of its Brillouin zone. No
assumptions are made on the size of the potential. We then prove the robustness
of such conical singularities to a restrictive class of perturbations, which
break the honeycomb lattice symmetry. General small perturbations of potentials
with Dirac points do not have Dirac points; their dispersion surfaces are
smooth. The presence of Dirac points in honeycomb structures is associated with
many novel electronic and optical properties of materials such as graphene.Comment: To appear in Journal of the American Mathematical Society; 54 pages,
2 figures [note: earlier replacement was original version
Wave packets in Honeycomb Structures and Two-Dimensional Dirac Equations
In a recent article [10], the authors proved that the non-relativistic
Schr\"odinger operator with a generic honeycomb lattice potential has conical
(Dirac) points in its dispersion surfaces. These conical points occur for
quasi-momenta, which are located at the vertices of the Brillouin zone, a
regular hexagon. In this paper, we study the time-evolution of wave-packets,
which are spectrally concentrated near such conical points. We prove that the
large, but finite, time dynamics is governed by the two-dimensional Dirac
equations.Comment: 34 pages, 2 figure
Geometric Analysis of Bifurcation and Symmetry Breaking in a Gross-Pitaevskii equation
Gross-Pitaevskii and nonlinear Hartree equations are equations of nonlinear
Schroedinger type, which play an important role in the theory of Bose-Einstein
condensation. Recent results of Aschenbacher et. al. [AFGST] demonstrate, for a
class of 3- dimensional models, that for large boson number (squared L^2 norm),
N, the ground state does not have the symmetry properties as the ground state
at small N. We present a detailed global study of the symmetry breaking
bifurcation for a 1-dimensional model Gross-Pitaevskii equation, in which the
external potential (boson trap) is an attractive double-well, consisting of two
attractive Dirac delta functions concentrated at distinct points. Using
dynamical systems methods, we present a geometric analysis of the symmetry
breaking bifurcation of an asymmetric ground state and the exchange of
dynamical stability from the symmetric branch to the asymmetric branch at the
bifurcation point.Comment: 22 pages, 7 figure
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