In this work, we show that the unique bulk-edge correspondence can serve as a fingerprint for identifying topological excitonic insulators, with the $\mathrm{LiFe}X$ ($X=\mathrm{S}$, Se, and Te) family as a prototype. First-principles Bethe-Salpeter equation calculations reveal excitonic instabilities in these spin-orbit coupling quantum anomalous Hall insulators. Effective Hamiltonian analyses indicate that spontaneous exciton condensation does not disrupt the gapless edge state but reconstructs the bulk gap to be almost independent of the spin-orbit coupling strength. This change in the bulk-edge correspondence can be experimentally inspected by angle-resolved photoemission spectroscopy or electron compressibility measurements, providing observational evidence for the identification of topological excitonic insulators.

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• Bethe-Salpeter equation
• Excitons
• Quantum anomalous Hall effect
• Spin-orbit coupling
• Topological phases of matter
• First-principles calculations