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Mirror symmetric twisted trilayer graphene (tTLG) is composed of even paritytwisted bilayer graphene (tBLG)-like bands and odd parity Dirac-like bands.Here, we study the mirror-symmetric and mirror-asymmetric Hofstadter-Moir\'{e}(HM) fractal bands of tTLG. A novel quantum parity Hall state is identified inmirror-symmetric tTLG at experimentally accessible charge densities. Thismirror symmetry-protected topological phase exhibits simultaneous quantizedHall and longitudinal resistances. The effects of the displacement field on theHM fractal bands of tTLG and topological phase transitions are also studied.The application of an electric displacement field results in an emergent weaklydispersive band at the charge neutrality point for a range of twist angles.This zero-energy state resides in the middle layer. It is isolated from the HMspectrum by an energy gap that scales proportional to the applied displacementfield, making it a prime candidate to host correlated topological states.

Hofstadter-Moir\'{e} Butterfly in Twisted Trilayer Graphene