Flat band exciton and quantum metric

We discuss the excitons in flat band systems. In particular, we ask the question of what determines the size of an exciton, namely the separation between the electron and hole, for a flat band system. The usual Bohr radius vanishes due to the infinite effective mass. We point out that the quantum metric gives a lower bound on the size of excitons.

To demonstrate the role of quantum metric, we will discuss the excitons in systems of quantum Hall bilayer as well as a toy model of Lieb lattice bilayer. In the quantum Hall bilayer, the exciton size is given by the magnetic length as usually expected. Indeed, the magnetic length can be viewed as the quantum metric of the wavefunctions of the lowest Landau level. Meanwhile, in the toy model of Lieb lattice bilayer, the exciton size is generally larger and saturates to the quantum metric when the total momentum of exciton is zero. Given the fundamental role played by the quantum metric in exciton physics, potential experimental implications about excitonic phases will be discussed for materials with large quantum metric.