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Stable topological phase transitions without symmetry indications in NaZnSb$_{1-x}$Bi$_x$

We study topological phase transitions in tetragonal NaZnSb$_{1-x}$Bi$_x$, driven by the chemical composition $x$. Notably, we examine mirror Chern numbers that change without symmetry indicators. First-principles calculations are performed to show that NaZnSb$_{1-x}$Bi$_x$ experiences consecutive topological phase transitions, diagnosed by the strong $\mathbb Z_{2}$ topological index $ν_{0}$ and two mirror Chern numbers $μ_{x}$ and $μ_{xy}$. As the chemical composition $x$ increases, the topological invariants ($μ_{x}μ_{xy}ν_{0}$) change from $(000)$, $(020)$, $(220)$, to $(111)$ at $x$ = 0.15, 0.20, and 0.53, respectively. A simplified low-energy effective model is developed to examine the mirror Chern number changes, highlighting the central role of spectator Dirac fermions in avoiding symmetry indicators. Our findings suggest that NaZnSb$_{1-x}$Bi$_{x}$ can be an exciting testbed for the exploration of the interplay between the topology and symmetry.