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

Recent theoretical studies have introduced the concept of fragile topological insulators, whose nontrivial band structure can be trivialized by addition of a trivial band. Fragile topological insulators fall outside the scope of the K-theoretical classification that is concerned with stable topological phases, like Chern insulators. In this talk, we present theoretical models of three-dimensional rotation-symmetric fragile topological insulators in class AI, which have n surface Dirac cones that are protected by time-reversal and n-fold rotation (C_n) symmetries (n = 2, 4, 6). We clarify that the topological insulators are classified into two kinds of fragile topological insulators. One is a fragile Z topological insulator whose only nontrivial topological index is the Euler class that specifies the number of surface Dirac cones. The other is a fragile Z₂ topological insulator having gapless surface states with either a quadratic band touching or four (six) Dirac cones, which are protected by time-reversal and C₄ (C₆) symmetries. Furthermore, we demonstrate that the surface states are gapped out through hybridization with an s-orbital band localized at the surface.