Kibble-Zurek behavior in a topological phase transition with a quadratic band crossing

The Kibble-Zurek (KZ) mechanism describes the scaling behavior when driving a system across a continuous symmetry-breaking transition. Previous studies have shown that the KZ-like scaling behavior also lies in the topological transitions in the Qi-Wu-Zhang model (2D) and the Su-Schrieffer-Heeger model (1D), although symmetry breaking does not exist in these systems. Both models with linear band crossings yield $\ensuremath{\nu}\ensuremath{\simeq}1$ and $z\ensuremath{\simeq}1$. A natural question is whether different critical exponents can emerge in topological transitions beyond the linear band crossings. In this paper, we look into the KZ behavior in a topological 2D checkerboard lattice with a quadratic band crossing. We investigate from dual perspectives: momentum distribution of the Berry curvature in clean systems for simplicity, and real-space analysis of domain-like local Chern marker configurations in disordered systems, which is a more intuitive analog to conventional KZ description. In equilibrium, we find the correlation length diverges at the transition with a power $\ensuremath{\nu}\ensuremath{\simeq}1/2$. Then, by slowly quenching the system across the topological phase transition, we find that the freeze-out time ${t}_{\mathrm{f}}$ and the unfrozen length scale $\ensuremath{\xi}({t}_{\mathrm{f}})$ both satisfy the KZ scaling, verifying $z\ensuremath{\simeq}2$. We subsequently explore KZ behavior in topological phase transitions with other higher-order band crossings and find the relationship between the critical exponents and the order. Our results extend the understanding of the KZ mechanism and nonequilibrium topological phase transitions.