We study the BCS-BEC crossover phenomenon of triplet exciton condensation using a half-filled bilayer Hubbard model.We calculate the dynamical spin structure factor and the exciton wave function on the phase boundary between the antiferromagnetically ordered and disordered phases.In the BCS regime, the formation and condensation of particle-hole bound states, namely excitons, occur simultaneously at the phase transition point, and the exciton wave function is extended in real space.In the BEC regime, on the other hand, bound states are well defined over the whole Brillouin zone also in the disordered phase, and the size of particle-hole pairs is smaller than a lattice space.Quantum criticality of charge-spin-orbital entangled states can emerge from intermediate coupling materials in the crossover regime.
The condensation of spin-orbit-induced excitons in
t2g4 electronic systems is attracting considerable attention. At large Hubbard
U, antiferromagnetism was proposed to emerge from the Bose-Einstein Condensation (BEC) of triplons (
Jeff=1). Here, we show that even at intermediate
U regimes, the spin-orbit exciton condensation is possible leading also to staggered magnetic order. The canonical electron-hole excitations (excitons) transform into local triplon excitations at large
U, and this BEC strong coupling regime is smoothly connected to the intermediate
U excitonic insulator region. We solved the degenerate three-orbital Hubbard model with spin-orbit coupling ($\ensuremath{\lambda}$) in one dimension using the density matrix renormalization group, while in two dimensions we use the Hartree-Fock approximation (HFA). Employing these techniques, we provide the full $\ensuremath{\lambda}$ versus
U phase diagrams for both one- and two-dimensional lattices. Our main result is that at intermediate Hubbard
U, increasing $\ensuremath{\lambda}$ at fixed
U the system transitions from an incommensurate spin-density-wave metal to a Bardeen-Cooper-Schrieffer (BCS) excitonic insulator, with coherence length
rcoh of
O(a) and
O(10a) in
1d and
2d, respectively, with
a being the lattice spacing. Further increasing $\ensuremath{\lambda}$, the system eventually crosses over to the BEC limit (with ${r}_{\mathrm{coh}}\ensuremath{\ll}a$).
We investigate electron-hole pair condensation in electron bilayers described by the square-lattice extended Falicov-Kimball model. Using exact diagonalization and variational cluster approximation techniques we first calculate the anomalous Green’s function to clarify the character of the excitons in momentum space. We then evaluate the coherence length ξ (in unit of the lattice constant a) from the corresponding condensation amplitude and demonstrate the smooth crossover between a BCS state of weakly paired electrons and holes (ξ/a 1) and a BEC state of tightly bound excitons (ξ/a 1) as the Coulomb attraction increases. Overcoming the finite-size effects of exact diagonalization while still taking into account the essential correlation effects we show that the variational cluster approximation provides an advantageous description of exciton condensation in strongly correlated electron systems.
Motivated by recent experimental and theoretical studies of excitonic insulators, we use the variational cluster approximation to calculate the single-particle Green's function and anomalous Green's function of the Falicov-Kimball model extended by including a finite valence bandwidth. We thus determine the single-particle excitation gap due to the bound-state formation of an electron and a hole. We also evaluate the order parameter of the excitonic insulating state indicating quantum coherence of condensed excitons. We thereby discuss the excitonic insulator that typifies either a BCS condensate of electron-hole pairs (weak-coupling regime) or a Bose-Einstein condensate of preformed excitons (strong-coupling regime). A BCS-BEC crossover of the model thus manifests itself as a function of the Coulombic coupling strength.
The crossover between low and high density regimes of exciton-polariton condensates is examined using a BCS wave-function approach. Our approach is an extension of the BEC-BCS crossover theory for excitons, but includes a cavity photon field. The approach can describe both the low density limit, where the system can be described as a Bose-Einstein condensate (BEC) of exciton-polaritons, and the high density limit, where the system enters a photon-dominated regime. In contrast to the exciton BEC-BCS crossover where the system approaches an electron-hole plasma, the polariton high density limit has strongly correlated electron-hole pairs. At intermediate densities, there is a regime with BCS-like properties, with a peak at nonzero momentum of the singlet pair function. We calculate the expected photoluminescence and give several experimental signatures of the crossover.
Photoinduced dynamics in an excitonic insulator is studied theoretically by using a two-orbital Hubbard model on the square lattice where the excitonic phase in the ground state is characterized by the BCS-BEC crossover as a function of the interorbital Coulomb interaction. We consider the case where the order has a wave vector $\mathbit{Q}=(0,0)$ and photoexcitation is introduced by a dipole transition. Within the mean-field approximation, we show that the excitonic order can be enhanced by the photoexcitation when the system is initially in the BEC regime of the excitonic phase, whereas it is reduced if the system is initially in the BCS regime. The origin of this difference is discussed from behaviors of momentum distribution functions and momentum-dependent excitonic pair condensation. In particular, we show that the phases of the excitonic pair condensation have an important role in determining whether the excitonic order is enhanced or not.
The BCS to Bose-Einstein condensation (BEC) crossover of electron-hole $(e\text{\ensuremath{-}}h)$ pairs in optically excited semiconductors is studied using the two-band Hubbard model with both repulsive and attractive interactions. Applying the self-consistent
t-matrix approximation combined with a local approximation, we examine the properties of a normal phase and an excitonic instability. The transition temperature from the normal phase to an $e\text{\ensuremath{-}}h$ pair condensed one is studied to clarify the crossover from an $e\text{\ensuremath{-}}h$ BCS-like state to an excitonic Bose-Einstein condensation, which takes place on increasing the $e\text{\ensuremath{-}}h$ attraction strength. To investigate effects of the repulsive interaction and the $e\text{\ensuremath{-}}h$ mass difference, we calculate the transition temperature for various parameters of the interaction strengths, the $e\text{\ensuremath{-}}h$ particle density, and the mass difference. While the transition temperature in the $e\text{\ensuremath{-}}h$ BCS regime is sufficiently suppressed by the repulsive interaction, that of the excitonic BEC is largely insensitive to it. We also show quantitatively that in the whole regime the mass difference leads to large suppression of the transition temperature.