Fragile topological insulators protected by rotation symmetry without spin-orbit coupling

We present a series of models three-dimensional rotation-symmetric fragile topological insulators in class AI (time-reversal symmetric and spin-orbit-free systems), which have gapless surface states protected by time-reversal ($T$) $n$-fold rotation ($C_n$) symmetries ($n=2,4,6$). Our are generalizations Fu's model spinless crystalline insulator, orbital degrees freedom play the role pseudo-spins. consider minimal Hamiltonian with $C_n$ symmetry discuss possible symmetry-protected states, i.e., quadratic band touching multiple Dirac cones linear dispersion. characterize structure bulk wave functions terms two kinds invariants obtained from Wilson loops: $\mathbb{Z}_2$ ($n=4,6$) symmetries, $C_2T$-symmetry-protected $\mathbb{Z}$ (the Euler class) when number occupied bands is two. Accordingly, our realize insulators. One insulator whose only nontrivial index that specifies cones. The other having either or four (six) cones, $C_4$ ($C_6$) symmetries. Finally, we instability against addition $s$-orbital demonstrate gapped out through hybridization surface-localized bands.