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The effect of spatially modulated interaction on quantum phase transition inone-dimensional interacting spinless fermion system is theoreticallyinvestigated by exact diagonalization and density matrix renormalization groupmethod. Our calculations show that the periodically modulated interaction candrive the spinless fermion system into topological charge density wave state.The topological state is encoded by quasiparticle end states and the fractionalquantized $e/2$ end charges, and characterized by Berry phase and Chern number.The quasiparticle energy spectra as a function of modulated interaction periodappears a stunning fractal-like structure. For the quasi-periodic case, thetopological phase transition can also occur. In a word, the spatially modulatedinteraction can be a new simple and elegant avenue towards realizing tointeracting topological phases.

Topological end states in a one-dimensional spatially modulated interaction spinless fermion model