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We propose that Kibble-Zurek scaling can be studied in optical lattices bycreating geometries that support, Dirac, Semi-Dirac and Quadratic BandCrossings. On a Honeycomb lattice with fermions, as a staggered on-sitepotential 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 theKibble-Zurek relation. We generalize the result for a passage through asemi-Dirac point in $d$ dimensions, in which spectrum is linear in $m$ paralleldirections and quadratic in rest of the perpendicular $(d-m)$ directions. Wefind 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 correlationlength exponents along the parallel and perpendicular directions, respectively.The scaling relations are also generalized to the case of non-linear quenching.

Quenching through Dirac and semi-Dirac points in optical Lattices: Kibble-Zurek scaling for anisotropic Quantum-Critical systems