Roles of Hund's rule coupling in excitonic density-wave states

Excitonic density-wave states realized by the quantum condensation of electron-hole pairs (or excitons) are studied in the two-band Hubbard model with Hund's rule coupling and the pair hopping term. Using the variational cluster approximation, we calculate the grand potential of the system and demonstrate that Hund's rule coupling always stabilizes the excitonic spin-density-wave state and destabilizes the excitonic charge-density-wave state and that the pair hopping term enhances these effects. The characteristics of these excitonic density-wave states are discussed using the calculated single-particle spectral function, density of states, condensation amplitude, and pair coherence length. Implications of our results in the materials' aspects are also discussed.

Figure 1

(a) Calculated grand potentials for the excitonic CDW and SDW states as a function of the variational parameter ${\Delta }^{\prime }(={\Delta }_0^{\prime },{\Delta }_z^{\prime })$ at $J/t=J^{\prime }/t=0$, 0.25, 0.5, 0.75, and 1. ${\Omega }_0$ is the grand potential in the normal (semimetallic) state. The crosses and dots indicate the stationary points of the excitonic CDW and SDW states, respectively. (b) $J(=J^{\prime })$ dependence of the grand potential at the stationary point for the normal (or semimetallic), excitonic CDW, and SDW states. (c) Optimized values of the grand potentials in the presence ($J^{\prime }=J$) and absence ($J^{\prime }=0$) of the pair hopping term. (d) $J$ dependence of the order parameters of the excitonic CDW and SDW states in the presence ($J^{\prime }=J$) and absence ($J^{\prime }=0$) of the pair hopping term.

Figure 2

Single-particle spectral function $A(\mathit{k},\omega )$ and DOS $N_{\alpha }\left(\omega \right)$ calculated by CPT at $J/t=J^{\prime }/t=0.5$. We show the results for the excitonic CDW state (metastable) in (a) and (c) and for the excitonic SDW state (stable) in (b) and (d). In (c) and (d), the solid, dashed, and dotted lines indicate the $f$ orbital, $c$ orbital, and total DOSs, respectively. The artificial Lorentzian broadening of $\eta /t=0.15$ is used for $A(\mathit{k},\omega )$ and $\eta /t=0.05$ is used for $N_{\alpha }\left(\omega \right)$. The Fermi level is located at $\omega =0$.

Figure 3

Calculated DOSs for the (a) excitonic CDW state and (b) excitonic SDW state at $J/t=J^{\prime }/t=0.5$. Solid and dashed lines indicate the DOSs of the $A$ and $B$ sublattices, respectively. The Lorentzian broadening of $\eta /t=0.05$ is used. The vertical line indicates the Fermi level.

Figure 4

Condensation amplitude $F\left(\mathit{k}\right)[=F_0\left(\mathit{k}\right)$ or $F_z\left(\mathit{k}\right)]$ calculated by CPT. We show the results for (a) the CDW state at $J/t=J^{\prime }/t=1.0$ (metastable), (b) the CDW/SDW states at $J=J^{\prime }=0$ (degenerate), and (c) the SDW state at $J/t=J^{\prime }/t=1.0$ (stable).

Figure 5

Calculated pair coherence length $\xi$ in units of the lattice constant. $J(=J^{\prime })$ dependence of $\xi$ is shown for the spin-singlet (open circles) and spin-triplet (solid circles) exciton condensations. The inset shows the results in the absence of the pair hopping term $J^{\prime }=0$ (open and solid squares), which are compared with the results in the presence of the pair hopping term $J^{\prime }=J$ (open and solid circles).

Figure 6

(a) The number of the conduction-band electrons $\langle n_c\rangle$ (or the valence-band holes) as a function of $J/t$ in the normal state (or ${\Delta }^{\prime }=0$), which is obtained using the atomic-limit relation $U^{\prime }=U-2J$ with $U/t=5,D/t=2$, and $J^{\prime }=0$. (b) Calculated grand potentials of the excitonic CDW and SDW states as a function of the variational parameter ${\Delta }^{\prime }(={\Delta }_0^{\prime },{\Delta }_z^{\prime })$, which are obtained using the atomic-limit relation $U^{\prime }=U-2J$ with $U/t=5,J/t=J^{\prime }/t=0.5$, and $D/t=2$. The crosses and circles indicate the stationary points of the excitonic CDW and SDW states, respectively.