Quenching through Dirac and semi-Dirac points in optical lattices: Kibble-Zurek scaling for anisotropic quantum critical systems

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/τ, where τ 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 the rest of the perpendicular ( d - m ) directions. We find that the defect density is given by 1/τ m ν || z || +( d - m )ν ⊥ z ⊥ where ν || , z || and ν ⊥ , z ⊥ 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.