  We investigate the self-organization of strongly interacting particles
confined in 1D and 2D. We consider hardcore bosons in spinless Hubbard lattice
models with short-range interactions. We show that many-body states with
topological features emerge at different energy bands separated by large gaps.
The topology manifests in the way the particles organize in real space to form
states with different energy. Each of these states contains topological
defects/condensations whose Euler characteristic can be used as a topological
number to categorize states belonging to the same energy band. We provide
analytical formulas for this topological number and the full energy spectrum of
the system for both sparsely and densely filled systems. Furthermore, we
analyze the connection with the Gauss-Bonnet theorem of differential geometry,
by using the curvature generated in real space by the particle structures. Our
result is a demonstration of how states with topological characteristics,
emerge in strongly interacting many-body systems following simple underlying
rules, without considering the spin, long-range microscopic interactions, or
external fields.
