Exciton condensation in strongly correlated electron bilayers
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FIG. 13. (Color online) Magnetic modes of the exciton condensate phase. We have set t = V = 2 eV, J = 125 meV, and α = 0.04 and the exciton density is either ρ = 0.15 (left column) or ρ = 0.27 (right column). (a) and (b) Magnetic mode dispersions for a quantum paramagnet, where the excitation spectrum is governed by propagating triplet modes. These modes have a gap of order J⊥ and a bandwidth of order Jz. (c) and (d) Magnetic mode dispersions for the actual exciton t-J model. In contrast to the results of (a) and (b), the actual triplet modes have enhanced kinetics.38 The modes are split in a spin-dominated branch with small gap and large bandwidth proportional to the superfluid density and an exciton-dominated branch with a large gap and a small bandwidth. (e) and (f) The spin susceptibility as measured in, for example, RIXS. Here, the enhanced triplet mode is clearly visible. (g) and (h) The susceptibility associated with the operator E1m, which indicates that the upper spin branch is dominated by exciton physics. 
FIG. 1. (Color online) Side view of a strongly correlated electron bilayer with an exciton present. The red arrows denote the spin of the localized electrons, and the exciton is a bound state of a double occupied and an empty site. 
FIG. 3. (Color online) Results from the semiclassical Monte Carlo simulations. Here shown are color plots with the exciton density ρ on the horizontal axes and the hopping parameter t (in eV) on the vertical axes. Other parameters are fixed at J = 125 meV, α = 0.04, and V = 2 eV. The five measurements shown here are the Néel order parameter (35), the checkerboard order parameter (36), the superfluid density (37), the phase coherence (38), and the ratio signaling phase separation according to Eq. (40), 0 means complete phase separation, 1 means no phase separation. Notice that the prominent line at ρ = 0.5 signals the checkerboard phase. 
FIG. 6. (Color online) The canonical mean-field phase diagram for typical values of J = 125 meV, α = 0.04, and V = 2 eV while varying t and the exciton density ρ. In the absence of exciton, at ρ = 0, we have the pure antiferromagnetic Néel phase (AF). Exactly at half-filling of excitons (ρ = 1/2) and small hoping energy t < 2V , we find the checkerboard phase (CB) where one sublattice is filled with excitons and the other sublattice is filled with singlets. For large values of t , we find the singlet exciton condensate (EC), given by the wave function∏ 
FIG. 11. (Color online) The exciton modes in the antiferromagnetic phase in the antiadiabatic regime t J . Here, we have chosen t = 2 eV, J = 125 meV, and α = 0.04. (a) Exciton mode dispersions are, just like in Fig. 10, renormalized with respect to the free hard-core boson results. (b) The mode dispersion for free hard-core bosons on a lattice. (c) The susceptibility computed with the equations of motion method, following the dispersions in (a). (d) The susceptibility computed using the fully interacting LSWSCBA theory. Upon inclusion of the interactions, the susceptibility gets extremely renormalized with respect to (c). The large exciton kinetic energy together with the relatively spin dynamics create an effective potential for the exciton: the exciton becomes localized and the confinement generates a ladder spectrum. Note that, therefore, in the antiadiabatic regime, the free results (a) and (c) cannot be trusted. 
FIG. 12. (Color online) Dispersions and susceptibilities of the Goldstone mode associated with the exciton condensate. We have set t = V = 2 eV, J = 125 meV, and α = 0.04, and the exciton density is either ρ = 0.15 (left column) or ρ = 0.27 (right column). (a) and (b) Mode dispersions for hard-core bosons. In the simple hard-core boson model, the condensate phase clearly shows the superfluid phase mode, linear at small momenta. (c) and (d) Mode dispersions in the full t-J model. Here, the Goldstone mode has a similar dispersion as in the XXZ model of (a) and (b). The speed of the mode scales with the superfluid density. At higher densities, the mode softens around (π,π ), and when this gap closes, a first-order transition to the checkerboard phase sets in. (e) and (f) The absorptive part of the charge susceptibility, which can be measured with for example EELS or RIXS.
Citations
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