We apply gain/loss to honeycomb photonic lattices and show that the dispersion relation is identical to tachyons - particles with imaginary mass that travel faster than the speed of light. This is accompanied by PT-symmetry breaking in this structure. We further show that the PT-symmetry can be restored by deforming the lattice

We study the critical behavior of Anderson localized modes near intersecting flat and dispersive bands in the quasi-one-dimensional diamond ladder with weak diagonal disorder W . The localization length ξ of the flat band states scales with disorder as ξ ∼ W −γ , with γ ≈ 1.3, in contrast to the dispersive bands with γ = 2. A small fraction of dispersive modes mixed with the flat band states is responsible for the unusual scaling. Anderson localization is therefore controlled by two different length scales. Nonlinearity can produce qualitatively different wave spreading regimes, from enhanced expansion to resonant tunneling and self-trapping.

We study the scattering of waves off a potential step in deformed honeycomb lattices. For deformations below a critical value, perfect Klein tunneling is obtained; i.e., a potential step transmits waves at normal incidence with nonresonant unit-transmission probability. Beyond the critical deformation a gap forms in the spectrum, and a potential step perpendicular to the deformation direction reflects all normally incident waves, exhibiting a dramatic transition form unit transmission to total reflection. These phenomena are generic to honeycomb lattices and apply to electromagnetic waves in photonic lattices, quasiparticles in graphene, and cold atoms in optical lattices.

We study the phenomena associated with symmetry breaking in honeycomb photonic lattices. As the honeycomb structure is gradually deformed, conical diffraction around its diabolic points becomes elliptic and eventually no longer occurs. As the deformation is further increased, a gap opens between the first two bands, and the lattice can support a gap soliton. The existence of the gap soliton serves as a means to detect the symmetry breaking and provide an estimate of the size of the gap.

We study linear and nonlinear wave dynamics in the Lieb lattice, in the vicinity of an intersection point between two conical bands and a flat band. We define a pseudospin operator and derive a nonlinear equation for spin-1 waves, analogous to the spin-1/2 nonlinear Dirac equation. We then study the dynamics of wave packets that are associated with different pseudospin states, and find that they are distinguished by their linear and nonlinear conical diffraction patterns.