Excitonic instability at the spin-state transition in the two-band Hubbard model

Using linear response theory with the dynamical mean-field approximation we investigate the particle-hole instabilities of the two-band Hubbard model in the vicinity of the spin-state transition. Besides the previously reported high-spin–low-spin order we find an instability towards triplet excitonic condensate. We discuss the strong and weak-coupling limits of the model, in particular, a connection to the spinful hard-core bosons with a nearest-neighbor interaction. Possible realization in perovskite cobaltites is briefly discussed.

Figure 1

(Left) The conceptual phase diagram of the two-band Hubbard model for $U=4J$. The shaded area marks the parameter range visited while varying the band asymmetry $\zeta$ and crystal field $\Delta$. (Right) 1P spectral densities obtained at the points marked by stars (top panel corresponds to the upper star) at temperatures just above the leading $T_c$.

Figure 2

The typical $q$ dependence of the leading eigenvalues of the susceptibility matrix: spin longitudinal (red), OD (green), and OO (blue) in a system with a large band asymmetry $\zeta =0.22$, $\Delta =3.40$ at temperatures 773 K, 644 K, and 580 K (left to right).

Figure 3

(Left) Leading eigenvalues for equal bandwidths ($\zeta =1$) and $\Delta =3.40$ eV at 1160 K. The blue OO mode diverges faster than the green OD mode. (Right) Splitting of the OO mode from (a) due to added cross hopping $V_{1,2}=0.1\mathrm{eV}$. The leading mode (twofold degenerate) has the form $a_{\sigma }^{†}{b}_{-\sigma }+b_{\sigma }^{†}{a}_{-\sigma }$ with $\sigma =\uparrow ,\downarrow$.

Figure 4

(a) Representative dependencies of the instability temperatures on the crystal field $\Delta$: $T_{\mathrm{OD}}$ (squares) for $\zeta =0.28$ and $T_{\mathrm{OO}}$ (circles) for $\zeta =0.55$. The open square marks the position of the reentrant transition taken from Ref. [11]. The blue line marks the estimated position of ${\Delta }_c$. (b) The $T$ dependence of the inverse eigenvalues ${\chi }_{\mathrm{OO}}^{-1}$ (circles) and ${\chi }_{\mathrm{OD}}^{-1}$ (squares) of the susceptibility at selected values of $\Delta$. The parameters $\zeta =0.28$, $\Delta =3.44$ eV (blue) correspond to $\Delta \gtrsim {\Delta }_c$, where the OD instability already disappeared. For $\zeta =0.55$, $\Delta =3.38$ eV (black) the OD instability exists only in a finite interval of temperatures. In both cases the OO is the leading instability, which is physically realized.

Figure 5

Instability of the normal phase as a function of band asymmetry $\zeta$ for various CF parameters $\Delta$. Open circles denote the divergence $T_{\mathrm{OO}}$ of the OO mode; solid squares mark the divergence $T_{\mathrm{OD}}$ of the OD mode. The lines are guides for the eye. The dashed vertical lines mark the $\zeta$'s for which the $\Delta$ dependencies of $T_{\mathrm{OO}}$ and $T_{\mathrm{OD}}$ are shown in Fig. 4.

Figure 6

An example of various combinations of the hoppings with orbitals of $s$ and $p_z$ symmetry. (a) $t_{a,b}>0$, $V_{1,2}=0$, (b) $t_{a,b}>0$, $V_1=-V_2$, and (c) $t_a>0$, $t_b<0$, $V_1=-V_2$.