  Exciton-exciton interactions are key to understanding non-linear optical and
transport phenomena in van der Waals heterobilayers, which emerged as versatile
platforms to study correlated electronic states. We present a combined
theory-experiment study of excitonic many-body effects based on first-principle
band structures and Coulomb interaction matrix elements. Key to our approach is
the explicit treatment of the fermionic substructure of excitons and dynamical
screening effects for density-induced energy renormalization and dissipation.
We demonstrate that dipolar blue shifts are almost perfectly compensated by
many-body effects, mainly by screening-induced self-energy corrections.
Moreover, we identify a crossover between attractive and repulsive behavior at
elevated exciton densities. Theoretical findings are supported by experimental
studies of spectrally-narrow interlayer excitons in atomically-reconstructed,
hBN-encapsulated MoSe$_2$/WSe$_2$ heterobilayers. Both theory and experiment
show energy renormalization on a scale of a few meV even for high injection
densities in the vicinity of the Mott transition. Our results revise the
established picture of dipolar repulsion dominating exciton-exciton
interactions in van der Waals heterostructures and open up opportunities for
their external design.
