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A moir\'{e} system is formed when two periodic structures have a slightlymismatched period, resulting in unusual strongly correlated states in thepresence of particle-particle interactions. The periodic structures can arisefrom the intrinsic crystalline order and periodic external field. Weinvestigate a one-dimensional Hubbard models with periodic on-site potential ofperiod $n_{0}$, which is commensurate to the lattice constant. For large $%n_{0}$, exact solution demonstrates that there is a midgap flat band with zeroenergy in the absence of Hubbard interaction. Each moir\'{e} unit cellcontributes two zero energy levels to the flat band. In the presence of Hubbardinteraction, the midgap physics is demonstrated to be well described by auniform Hubbard chain, in which the effective hopping and on-site interactionstrength, can be controlled by the amplitude and period of the external field.Numerical simulations are performed to demonstrate the correlated behaviors inthe finite-sized moir\'{e} Hubbard system, including the existence of $\eta$-pairing state, and bound pair oscillation. This finding provides a method toenhance the correlated effect by a spatially periodic external field.

Flat band and $\eta $-pairing states in one-dimensional Moir\'{e} Hubbard model