Computation of exciton binding energies in exciton condensation

Exciton binding energies are fundamental to understanding excitonic materials, especially those with the potential for ground-state exciton condensation. However, these energies are typically defined with significant limitations in their consideration of electron correlation. Here we present a variational theory for computing exciton binding energies in ground-state exciton condensates in which we define the binding as the energy difference between fully correlated many-electron systems with $M$ and $M\ensuremath{-}1$ excitons, respectively. The $(M\ensuremath{-}1)$ system is obtained by adding a constraint to the ground-state energy minimization that removes an exciton while allowing all other electronic degrees of freedom to relax. We perform the energy minimizations with variational calculations of the two-electron reduced density matrix (2-RDM) in which the additional constraint is treated along with the $N$-representability conditions---necessary constraints for the 2-RDM to represent an $N$-electron system---by semidefinite programming. We demonstrate the theory first in the Lipkin model and then in several stacked organic and inorganic systems that exhibit the beginnings of exciton condensation. We find that in the Lipkin model the traditional exciton binding model overbinds relative to the constrained approach. This has significant implications for theoretical characterizations of exciton condensates which rely on exciton binding energy to make predictions regarding condensate stability and critical temperatures. This correlated approach to defining and computing exciton binding energies may therefore have important applications for understanding the relationship between binding and condensation, especially for the BCS-BEC crossover.