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 rest of the perpendicular $(d-m)$ directions. We find that the defect density is given by $1 /{τ^{mν_{||}z_{||}+(d-m)ν_{\perp}z_{\perp}}}$ where $ν_{||}, z_{||}$ and $ν_{\perp},z_{\perp}$ 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.

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