Ph.D Student | Bahat-Treidel Omri |
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Subject | K-Space Singularities in Nonlinear Wave Systems |

Department | Department of Physics |

Supervisor | ? 18? Mordechai Segev |

Full Thesis text |

In the following dissertation I describe my
study of linear and nonlinear wave dynamics in honeycomb lattices. As opposed
to most periodic systems, the band structure of the honeycomb lattice is
singular in two different points. In the vicinity of these singularities, the
energy manifold, *E**(**k**)*, is a cone known in
the literature as the Dirac cone, since the Bloch waves that reside in the vicinity of
these singularities are very well described by the massless Dirac Hamiltonian.
However, the dispersion relation alone does not capture the entire uniqueness
of the system: since the honeycomb lattice is not a Bravais lattice but rather
it is made of two trigonal sub lattices. Therefore, the Bloch waves of the
system have two components that describe the amplitude of the wave in each
sub-lattice. This additional degree of freedom is often referred to as
pseudo-spin. The Dirac equation describing the waves is written in the
pseudo-spin space, not the real spin, and is therefore relevant for describing
bosons such as electro-magnetic waves in periodic media and cold atoms in an
optical lattice. In my research, I put a great deal of emphasis on deformed
honeycomb lattice, i.e., breaking the *C*_{3}_{v} symmetry of the lattice. The symmetry can be broken simply
by applying strain to a graphene sheet, modifying the amplitude of one of the
beams that generate the optical lattice for cold atoms, or changing the
distance between waveguides. When the* C*_{3}* _{v}* symmetry is broken, the
Dirac points move toward each other, eventually annihilate each other and form
a gap. Under such circumstances, the effective Dirac Hamiltonian is no longer
valid and it is extremely anisotropic. In addition, when the two cones are not
distinct another degree of freedom (valley) is lost, and as a consequence some
of the ”missing” Hall plateaus should appear. I have focused on studying the
nonlinear evolution of wave-packets comprising of waves that reside close to
the Dirac points in deformed lattices. To be more specific, I have been
studying the modification of conical diffraction when nonlinearity and
deformations are applied to the lattice. In addition, I studied Klein tunneling
in deformed lattices in the Dirac approximation and beyond it in the both
linear and the nonlinear regime.