Title: Single-Spin Waved-Brim Flat-Top Hat in the Band Edge of GdIH Monolayer

Abstract: Exotic electronic bands, such as flat bands, linear crossing bands, spontaneously valley- or spin-polarized bands, in two-dimensional materials have been the hot topics in condensed matter physics. Herein, we first propose a general dispersion model for possible hat-like electronic bands, and then identify an intriguing single-spin \emph{waved-brim flat-top hat} in the valence band edge of a stable ferromagnetic semiconducting electrene (i.e., Janus GdIH monolayer), which can be well described by a simplified two-bands Hamiltonian model. Specifically, the hat-band has a waved brim with six valleys along the boundary of the first Brillouin zone; meanwhile it holds a flat top close to the Fermi level, resulting in the emergence of single-spin van Hove singularities divergence and Lifshitz transitions. Owing to the breaking of both time-reversal and space inversion symmetries, a sizable spontaneous valley polarization is formed between the adjacent brim valleys, which provides the opportunity to realize the high-temperature anomalous valley Hall effect. Particularly, via modest strains and carriers doping, various conductive bipolar-states (spin-up vs. spin-down, K valley vs. $-$K valley, and ultra-low-speed vs. ultra-high-speed) can be modulated out from the distorted waved-brim flat-top hat of GdIH ML.