Quantum-metric-enabled exciton condensate in double twisted bilayer graphene

Flat-band systems are a promising platform for realizing exotic collective ground states with spontaneously broken symmetry because the electron-electron interactions dominate over the kinetic energy. A collective ground state of particular interest is the chased after exciton condensate (EC). However, in flat band systems other collective ground states can compete with an EC phase, and the conventional treatment of the effect of thermal and quantum fluctuations predicts the EC phase should be unstable. Here, using double twisted bilayer graphene (TBLG) heterostructures as an example, we show that for realistic interaction strengths the EC phase is favored with respect to other TBLG's phases -- orbital magnetism and superconductivity -- when the TBLGs have opposite doping, and that the quantum metric of the Bloch wave functions stabilizes the EC, reversing the conclusion that would be drawn from the conventional approach in which quantum metric contributions are neglected. Our results suggest that the quantum metric plays a critical role in determining the stability of exciton condensates in double layers formed by systems with flat-bands.