Slave Rotor Approach to Exciton Condensation in a Two-band System

Slave rotor mean field (SRMF) theory is employed to study exciton formation in the extended Falicov-Kimball Model (EFKM). In this theory, charge and spin (or orbital) degrees are treated as independent degrees of freedom, coupled by a constraint. Using this formalism in single as well as cluster extension, we capture the effective many body scales beyond conventional mean-field theory. While the formation of exciton is favored by the interband hybridization $V$, it is strongly influenced by the on-site Coulomb interaction $U$. Beyond a critical hybridization, there is a condensation of exciton, effectively giving rise to a crossover from metal to an excitonic insulator phase. The system goes from a metal to an excitonic insulating state, with a first order jump in the excitonic order parameter. Moreover, the behavior of excitonic averages differs from earlier results from Hartree-Fock mean-field theory. Low-$U$ results show that excitonic order parameter ($\Delta$) is continuous across the transition both for single as well as two-site approximation. The transition changes to weakly first order in the intermediate $U$ for cluster case. On the other hand, large $U$ limit shows a continuous transition for cluster but remains first order in the single-site approximation. The slave rotor also indicates an excitonic metallic region in both single and cluster cases, while there is an orbital liquid in the insulating regime in the cluster theory. The $\Delta-V$ graph shows step-like behavior when both the bands have the same parity. For cluster approximation, a second order to first order transition in $\Delta-V$ is obtained by tuning the hopping parameter of the localized band.