We determine the phase diagram of excitons in a symmetric transition-metal dichalcogenide 3-layer heterostructure. First principles calculations reveal interlayer exciton states of a symmetric quadrupole, from which higher energy asymmetric dipole states are composed. We find quantum phase transitions between a repulsive quadrupolar and an attractive staggered dipolar lattice phases, driven by a competition between interactions and single exciton energies. The different internal quantum state of excitons in each phase is a striking example of a system where single-particle and interacting many-body states are coupled.
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