  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 here. Both models with linear
band crossings give that $\nu=1$ and $z=1$. We wonder whether different
critical exponents can be acquired in topological transitions beyond linear
band crossing. In this work, 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 with a power $\nu\simeq 1/2$. 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 $\xi(t_\mathrm{f})$ both satisfy
the KZ scaling, verifying $z\simeq 2$. We subsequently explore KZ behavior in
topological phase transitions with other higher-order band crossing and find
the relationship between the critical exponents and the order. Our results
extend the understanding of the KZ mechanism and non-equilibrium topological
phase transitions.
