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The occupation of more than one single-particle state and hence the emergenceof fragmentation is a many-body phenomenon universal to systems of spatiallyconfined interacting bosons. In the present study, we investigate the effect ofthe range of the interparticle interactions on the fragmentation degree of one-and two-dimensional systems. We solve the full many-body Schr\"odinger equationof the system using the recursive implementation of the multiconfigurationaltime-dependent Hartree for bosons method, R-MCTDHB. The dependence of thedegree of fragmentation on dimensionality, particle number, areal or linedensity and interaction strength is assessed. It is found that for contactinteractions, the fragmentation is essentially density independent in twodimensions. However, fragmentation increasingly depends on density the morelong-ranged the interactions become. The degree of fragmentation is increasing,keeping the particle number $N$ fixed, when the density is decreasing asexpected in one spatial dimension. We demonstrate that this remains,nontrivially, true also for long-range interactions in two spatial dimensions.We, finally, find that within our fully self-consistent approach, thefragmentation degree, to a good approximation, decreases universally as$N^{-1/2}$ when only $N$ is varied.

Condensate fragmentation as a sensitive measure of the quantum many-body behavior of bosons with long-range interactions