  We study the quantum self-organization of interacting particles in
one-dimensional(1D) many-body systems, modeled via Hubbard chains with
short-range interactions between the particles. We show the emergence of 1D
states with density-wave and clustering order, related to topology, at odd
denominator fillings that appear also in the fractional-quantum-Hall-effect
(FQHE), which is a 2D electronic system with Coulomb interactions between the
electrons and a perpendicular magnetic field. For our analysis we use an
effective topological measure applied on the real space wavefunction of the
system, the Euler characteristic describing the clustering of the interacting
particles. The source of the observed effect is the spatial constraints imposed
by the interaction between the particles. In overall, we demonstrate a simple
mechanism to reproduce many of the effects appearing in the FQHE, without
requiring a Coulomb interaction between the particles or the application of an
external magnetic field.
