  We determine the zero temeperature phase diagram of excitons in the symmetric
transition-metal dichalcogenide tri-layer heterosctructure WSe2/MoSe2/WSe2.
First principle calculations reveal two distinct types of interlayer excitonic
states, a lower energy symmetric quadrupole and a higher energy asymmetric
dipole. While interaction between quadrupolar excitons is always repulsive,
anti-parallel dipolar excitons attract at large distances. We find quantum
phase transitions between a repulsive quadrupole lattice phase and a staggered
(anti-parallel) dipolar lattice phase, driven by the competition between the
exciton-exciton interactions and the single exciton energies. Remarkably, the
intrinsic nature of each interlayer exciton is completely different in each
phase. This is a striking example for the possible rich quantum physics in a
system where the single particle properties and the many-body state are
dynamically coupled through the particle interactions.
