Kibble-Zurek scaling immune to anti-Kibble-Zurek behavior in driven open systems at the limit of loss difference

We investigate the dissipative quench dynamics in a family of two-band fermionic systems by linearly ramping the staggered on-site energy. In the Lindblad formalism, we present an analytical solution in the presence of uniform loss or loss difference on bipartite lattices, which tells that dissipation exponentially suppresses the Kibble-Zurek (KZ) scaling behavior and the quantum jump term of the dissipation is responsible for the anti-KZ (AKZ) behavior. Interestingly, we find two different scaling behaviors at the limit of loss difference. Both scaling behaviors arise from the gapless Liouvillian. But one is accompanied by impulse stage rendered by the criticality of the system, so that it is ascribed to the universal KZ scaling law. Another depends on the dissipation strength and there is no impulse stage in it. We also point out a convenient way to observe the two new scaling behaviors by counting the number of residual particles in the end, since it is immune to the influence of AKZ behavior. We illustrate our findings through the prototypical one-dimensional Rice-Mele model first. Then, in the one-dimensional Shockley model and the two-dimensional Haldane model for Chern insulators, we show that the two scaling behaviors can appear together or separately with appropriate quench protocols.