Exact-diagonalization study of exciton condensation in electron bilayers

We report on small-cluster exact-diagonalization calculations which prove the formation of electron-hole pairs (excitons) as a prerequisite for spontaneous interlayer phase coherence in double-layer systems described by the extended Falicov-Kimball model. Evaluating the anomalous Green's function and momentum distribution function of the pairs, and thereby analyzing the dependence of the exciton binding energy, condensation amplitude, and coherence length on the Coulomb interaction strength, we demonstrate a crossover between a BCS-like electron-hole pairing transition and a Bose-Einstein condensation of tightly bound preformed excitons. We furthermore show that a mass imbalance between electrons and holes tends to suppress the condensation of excitons.

Figure 1

(a) Schematic representation of the DL EFKM cluster model with $N_s=16$ sites (32 orbitals). (b) Noninteracting tight-binding band structure and (c) square lattice Brillouin zone. Dots indicate the allowed momenta of the $4\times 4$ lattice with periodic boundary conditions. Throughout this work, we assume filling factors $n^f=0.75$ and $n^c=0.25$, i.e., $(N_f,N_c)=(12,4)$, which means $n^h=n^e=0.25$, irrespective of $U$. The red and blue lines in (c) show the perfectly matching hole and electron Fermi surfaces, respectively, with finite-lattice Fermi momenta ${\mathit{k}}_{\mathrm{F}}$ located at $\mathit{k}=(\pm \pi /2,0)$ and $(0,\pm \pi /2)$.

Figure 2

Condensation amplitude $F_{\mathit{k}}$ (upper panels) and momentum distribution function $N_{\mathit{q}}$ of excitons (lower panels) in the mass-symmetric DL EFKM with $U/t=0.5$ (left), 5 (middle), and 50 (right).

Figure 3

Coherence length $\xi$ (squares), order parameter $\Delta$ (diamonds), and exciton binding energy $|E_B|$ (circles) for the mass-symmetric DL EFKM as functions of $U/t$. For comparison, the asymptotics in the strong-coupling limit $\Delta \propto 0.45U$ (dashed line) and $|E_B|\propto U$ (dotted line) have been inserted.

Figure 4

Binding energy $|E_B|/U$ (left ordinate) and coherence length $\xi$ (right ordinate) for the mass-asymmetric (filled symbols) and mass-symmetric (open symbols) DL EFKM as functions of $|t_f|/U$ at $t_c/U=1$.

Figure 5

Exciton binding energy $E_B/U$ (left panel) and coherence length $\xi$ (right panel) of the DL EFKM in the $t_c/U$-$|t_f|/U$ plane.