Exciton-Exciton Interactions: A Variational and Path Integral Study

Abstract

Currently, two-dimensional materials wherein bound electron-hole pairs, known as excitons, play the predominant role in the material's dynamics, are receiving a great deal of attention due to their potential applicability for optoelectronic devices. The properties of single excitons in these systems are well understood; however, a general description of the interactions between excitons is lacking. Prior attempts at studying these interactions tend to neglect the composite nature of the exciton, indirectly omitting certain particle exchanges that can occur in a system of many excitons.

In this work, we study interactions between Wannier excitons using a variational approach, to obtain an effective exciton-exciton potential between two ground-state excitons. When considered for hydrogen-like excitons with a heavy hole, the potential reproduces the Heitler-London approach for two interacting hydrogen atoms.

Additionally, we develop a many-body theory for a gas of excitons, using the path-integral formalism; resulting in an excitonic action that, formally, includes $N$-body interactions between excitons. From that, we derive an effective action that includes a two-body interaction, which aligns with the results of the variational principle when evaluated on-shell.

The obtained results lay a general groundwork for the investigation of exciton-exciton interactions which can, in principle, be used for the study of biexciton formation and the Bose-Einstein condensation of excitons.