Quantum condensation in electron–hole systems: excitonic BEC–BCS crossover and biexciton crystallization

Quantum condensation of electron-hole (e-h) systems in photoexcited semiconductors is reviewed from a theoretical viewpoint, stressing the exciton Bose-Einstein condensation (BEC), the e-h BCS-type condensed state, the exciton Mott transition, and the biexciton crystallization. First, we discuss the crossover between the exciton BEC and the e-h BCS states at low temperature using the self-consistent t-matrix and local approximations, applied to the high-dimensional two-band Hubbard model with both repulsive and attractive on-site interactions. We also study the metal-insulator transition (called the 'exciton Mott transition') at zero and finite temperatures, investigated with the dynamical mean-field theory. Away from half-filling we find excitonic/biexcitonic insulating phases and the first-order transition between metallic and insulating states. Second, in a one-dimensional e-h system, we employ the exciton bosonization and renormalization-group techniques to clarify quantum orders at zero temperature. The most probable ground state exhibits the biexciton crystallization, which reflects the Tomonaga-Luttinger liquid properties, the e-h backward scattering, and the long-range Coulomb interaction. The one-dimensional e-h system is insulating even at the high-density limit, hence the exciton Mott transition never occurs at zero temperature in one dimension.