  By using the variational matrix product state method, we numerically study
the interacting Kitaev chain with spatially varying periodic and quasi-periodic
potentials and the latter follows the Fibonacci sequence. The edge correlation
functions of Majorana fermions and low-lying ground states are computed to
explore the robustness of topological superconducting phase. It is found that
the original topological nontrivial phase is separated into to two branches by
an emergent topological trivial phase, as a result of the competition among
spatially varying potential, electronic Coulomb interaction and chemical
potential. The analysis of energy gap and occupation number together suggests
that the spontaneous symmetry breaking and the lift of degeneracy in the
topological trivial phase are enabled by a potential-induced Fracton mechanism,
namely the pairing of four Majorana fermions. It can be broken by further
enhancing the interaction, and then the nontrivial phase reemerges. The
evolution from the emergent fractal structure of population outside the
critical region to the original structure of charge density wave is
investigated as well.
