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/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 the rest of the perpendicular (d-m) directions. We find that the defect density is given by 1/tau(mv+vertical bar vertical bar(d-m)v perpendicular to z perpendicular to) where v parallel to, z parallel to and v(perpendicular to), z(perpendicular to) 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. Copyright (C) EPLA, 2010