  We propose that Kibble-Zurek scaling can be studied in optical lattices by
creating geometries that support, Dirac, Semi-Dirac and Quadratic Band
Crossings. On a Honeycomb lattice with fermions, as a staggered on-site
potential is varied through zero, the system crosses the gapless Dirac points,
and we show that the density of defects created scales as $1/\tau$, where
$\tau$ is the inverse rate of change of the potential, in agreement with the
Kibble-Zurek relation. We generalize the result for a passage through a
semi-Dirac point in $d$ dimensions, in which spectrum is linear in $m$ parallel
directions and quadratic in rest of the perpendicular $(d-m)$ directions. We
find that the defect density is given by $ 1
/{\tau^{m\nu_{||}z_{||}+(d-m)\nu_{\perp}z_{\perp}}}$ where $\nu_{||}, z_{||}$
and $\nu_{\perp},z_{\perp}$ are the dynamical exponents and the correlation
length exponents along the parallel and perpendicular directions, respectively.
The scaling relations are also generalized to the case of non-linear quenching.
