Competition between excitonic charge and spin density waves: Influence of electron-phonon and Hund's rule couplings

We analyze the stability of excitonic ground states in the two-band Hubbard model with additional electron-phonon and Hund's rule couplings using a combination of mean-field and variational cluster approaches. We show that both the interband Coulomb interaction and the electron-phonon interaction will cooperatively stabilize a charge density wave (CDW) state which typifies an “excitonic” CDW if predominantly triggered by the effective interorbital electron-hole attraction or a “phononic” CDW if mostly caused by the coupling to the lattice degrees of freedom. By contrast, the Hund's rule coupling promotes an excitonic spin density wave. We determine the transition between excitonic charge and spin density waves and comment on a fixation of the phase of the excitonic order parameter that would prevent the formation of a superfluid condensate of excitons. The implications for exciton condensation in several material classes with strongly correlated electrons are discussed.

Figure 1

(a) Grand potential $\mathrm{\Omega }$ as a function of the variational parameters ${\mathrm{\Delta }}_0^{\prime }$ and ${\mathrm{\Delta }}_p$ for $U^{\prime }/t=4$ and $\lambda /t=0.15$. (b) ${\theta }_c$ dependence of $\mathrm{\Omega }$ taken the values of ${\mathrm{\Delta }}_0^{\prime }$ and ${\mathrm{\Delta }}_p$ optimized at ${\theta }_c=0$. Green dots indicate the stationary points.

Figure 2

(a) Grand potential $\mathrm{\Omega }$ as a function of the variational parameter ${\mathrm{\Delta }}_s^{\prime }$ at $U^{\prime }/t=4$ and $J/t=J^{\prime }/t=1$. (b) ${\theta }_s$ dependence of $\mathrm{\Omega }$ obtained using the value of ${\mathrm{\Delta }}_s^{\prime }$ optimized at ${\theta }_s=0$. Dots mark stable stationary points.

Figure 3

Optimized values of the grand potential ${\mathrm{\Omega }}_{\mathrm{opt}}$ in dependence on (a) $U^{\prime }/t$ and (b) $\lambda /t$. Here, ${\mathrm{\Omega }}_0$ is the grand potential in the normal (semimetallic) state. Order parameter ${\mathrm{\Phi }}_c$ for the excitonic CDW state as a function of (c) $U^{\prime }/t$ and (d) $\lambda /t$.

Figure 4

(a) Excitonic gap parameter ${\mathrm{\Delta }}_0$ (solid line) and phononic gap parameter ${\mathrm{\Delta }}_p$ (dashed line) divided by the total gap ${\mathrm{\Delta }}_c={\mathrm{\Delta }}_0+{\mathrm{\Delta }}_p$. ${\mathrm{\Delta }}_c, {\mathrm{\Delta }}_0$, and ${\mathrm{\Delta }}_p$ are separately plotted as a function of (b),(c) $U^{\prime }/t$ and (d),(e) $\lambda /t$.

Figure 5

$J$ dependence of the grand potential ${\mathrm{\Omega }}_{\mathrm{opt}}$ and the order parameter $\mathrm{\Phi }$ in the excitonic CDW (symbols) and SDW (solid line) states (a),(c) with ($J^{\prime }=J$) and (b),(d) without ($J^{\prime }=0$) the pair-hopping term, where $U^{\prime }/t=2.4$. ${\mathrm{\Omega }}_0$ is the grand potential of the normal semimetallic state.

Figure 6

Ground-state phase diagram of the two-band Hubbard model with electron-phonon and Hund's rule couplings showing the stability regions of excitonic CDW and SDW phases. Results are obtained, in the presence ($J^{\prime }=J$) and absence ($J^{\prime }=0$) of the pair-hopping term, by combining the mean-field and VCA approaches, for a two-dimensional (square) lattice at half filling, where $U^{\prime }/t=2.4$.