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

TL;DR

This paper investigates Kibble-Zurek scaling in optical lattices with Dirac, semi-Dirac, and quadratic band crossings, deriving defect density scaling laws for anisotropic quantum-critical systems during parameter quenches.

Contribution

It introduces a framework for studying Kibble-Zurek scaling in anisotropic systems with Dirac and semi-Dirac points, including generalized defect scaling laws for different dimensionalities.

Findings

Defect density scales as 1/τ for Dirac points.

Generalized defect scaling law for semi-Dirac points in d dimensions.

Scaling relations extended to non-linear quenching processes.

Abstract

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.