Competing phases and orbital-selective behaviors in the two-orbital Hubbard-Holstein model

We study the interplay between the electron-electron (e-e) and the electron-phonon (e-ph) interactions in the two-orbital Hubbard-Holstein model at half-filling using the dynamical mean-field theory. We find that the e-ph interaction, even at weak couplings, strongly modifies the phase diagram of this model and introduces an orbital-selective Peierls insulating phase (OSPI) that is analogous to the widely studied orbital-selective Mott phase (OSMP). At small e-e and e-ph couplings, we find a competition between the OSMP and the OSPI, while at large couplings, a competition occurs between Mott and charge-density-wave (CDW) insulating phases. We further demonstrate that the Hund's coupling influences the OSPI transition by lowering the energy associated with the CDW. Our results explicitly show that one must be cautious when neglecting the e-ph interaction in multiorbital systems, where multiple electronic interactions create states that are readily influenced by perturbing interactions.

Figure 1

The phase diagram for the two-orbital Hubbard-Holstein model in the e-ph interaction strength ($\lambda$)—Hubbard $U$ plane at charge density $n=2$ and temperature $\beta =200/W$. (a) and (b) show density plots of quasiparticle weights $Z_1$ and $Z_2$ on a logarithmic scale, respectively. The different phases are labeled as follows: metal (M), orbital-selective Mott phase (OSMP), Mott insulater (MI), charge-density wave (CDW), and orbital-selective Peierls insulator (OSPI). The white dots indicate points where the calculations were performed, and we plotted them to show phase boundaries. The color scale is plotted using a linear interpolation.

Figure 2

The quasiparticle weights (a) $Z_1$ and (b) $Z_2$ as a function of the e-ph interaction strength ($\lambda$) at different Hubbard $U$ values. Mean values of the local magnetic moments $m_{1z}^2, m_{2z}^2$ and phonon numbers ($N_{\mathit{ph}}$) are shown in (c), (d), and (e), respectively.

Figure 3

(a) The quasiparticle weight $Z_{\gamma }$ as a function of $U$ at a fixed $\lambda =0$. (b) The quasiparticle weight $Z_{\gamma }$ as a function of $\lambda$ at a fixed $U=0$. $Z_{\gamma }$ are results at $T=0.002\mathrm{eV}$ and ${\overline{Z}}_{\gamma }$ are results at $T=0.01\mathrm{eV}$. The solid lines and the dashed lines are results of increasing and decreasing $U$ or $\lambda$, respectively.

Figure 4

The phase diagram in the $\lambda -J/U$ plane at filling $n=2$. (a) and (b) plot quasiparticle weights $Z_1$ and $Z_2$, respectively. The labels used in this graph are the same as in Fig. 1. The Coulomb interaction is fixed at $U=0.8\mathrm{eV}$ and $U^{\prime }=U-2J$. The white dots indicate points where the calculations were performed, and we plotted them to show phase boundaries. The color scale is plotted using a linear interpolation.