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By means of the first-principles calculations and magnetic topologicalquantum chemistry, we demonstrate that the low energy physics in thecheckerboard antiferromagnetic (AFM) monolayer FeSe, very close to an AFMtopological insulator that hosts robust edge states, can be well captured by adouble-degenerate fragile topologically flat band just below the Fermi level.The Wilson loop calculations identify that such fragile topology is protectedby the $S_{4z}$ symmetry, which gives rise to an AFM higher-order topologicalinsulator that support the bound state with fractional charge $e/2$ at thesample corner. This is the first reported $S_{4z}$-protected fragiletopological material, which provides a new platform to study the intriguingproperties of magnetic fragile topological electronic states. Previousobservations of the edge states and bound states in checkerboard AFM monolayerFeSe can also be well understood in our work.

Fragile topologically flat band in the checkerboard antiferromagnetic monolayer FeSe