Nonlocal correlations in the orbital selective Mott phase of a one-dimensional multiorbital Hubbard model

We study nonlocal correlations in a three-orbital Hubbard model defined on an extended one-dimensional chain using determinant quantum Monte Carlo and density matrix renormalization group methods. We focus on a parameter regime with robust Hund's coupling, which produces an orbital selective Mott phase (OSMP) at intermediate values of the Hubbard $U$, as well as an orbitally ordered ferromagnetic insulating state at stronger coupling. An examination of the orbital- and spin-correlation functions indicates that the orbital ordering occurs before the onset of magnetic correlations in this parameter regime as a function of temperature. In the OSMP, we find that the self-energy for the itinerant electrons is momentum dependent, indicating a degree of nonlocal correlations while the localized electrons have largely momentum independent self-energies. These nonlocal correlations also produce relative shifts of the holelike and electronlike bands within our model. The overall momentum dependence of these quantities is strongly suppressed in the orbitally ordered insulating phase.

Figure 1

Fat band plot of the noninteracting band structure at a total filling of $\langle \widehat{n}\rangle =4$, where the thickness of the lines indicates the majority orbital content of the band. The topmost band has the narrowest bandwidth and is primarily of orbital 3 character. The lower two bands disperse over a much larger energy range and are primarily composed of orbitals 1 and 2, respectively.

Figure 2

Orbitally resolved electronic properties for $U/W=0.8$ $(W=2.45$ eV) at different temperatures. (a) The temperature dependence of orbital occupations. (b) The orbital resolved quasiparticle residue $Z_{\gamma }(k,\mathrm{i}\pi /\beta )$ at an inverse temperature $\beta =19.6/W$. (c) The normalized electron self energies $\mathrm{Im}{\mathrm{\Sigma }}_{\gamma }(k,\mathrm{i}\pi /\beta )$ at ${\omega }_n=\pi /\beta$ as a function of momentum. Each curve is normalized by its $k=0$ value to highlight the overall momentum dependence. The scale is determined by $\mathrm{Im}{\mathrm{\Sigma }}_{\gamma }(0,\mathrm{i}\pi /\beta )=-0.53,-0.57$, and $-2.53$ for $\gamma =1,2,3$, respectively, and in units of the bandwidth $W$. The blue, red, and green dash lines in (b) and (c) correspond to the bare Fermi momentum of the noninteracting bands. Panel (d) shows orbitally resolved quasiparticle residues $Z_{\gamma }(k_{\mathrm{F}}^0,\mathrm{i}\pi /\beta )$ and self-energies $\mathrm{Im}{\mathrm{\Sigma }}_{\gamma }(k_{\mathrm{F}}^0,\mathrm{i}\pi /\beta )$ at Fermi momentum as a function of temperature. In each panel, error bars smaller than the marker size have been suppressed for clarity.

Figure 3

Momentum dependence of Green functions $G(k,\tau =\beta /2)$ for (a) $U/W=0.1$, (b) 0.8, and (c) 2.0. The inverse temperature in all three cases is $\beta =19.6/W$. The blue, red, and green dash lines in each panel indicate the Fermi momentum of the three noninteracting bands. (d) $G(k_{\mathrm{F}},\tau =\beta /2)$ as a function of inverse temperatures $\beta$ for the OSMP $U/W=0.8$. Error bars smaller than the marker size have been suppressed for clarity.

Figure 4

Momentum dependence of the number operator $n_{\gamma }\left(k\right)=\frac{1}{2}{\sum }_{\sigma }\langle c_{k,\gamma ,\sigma }^{†}{c}_{k,\gamma ,\sigma }\rangle$ for each band. Results are shown for the noninteracting case $U=0$ (black dashed, $\square )$, $U/W=0.1$ (blue solid, $△)$, $U/W=0.8$ (red solid $\circ )$, and $U/W=2$ (green solid $\diamond )$ and at an inverse temperature of $\beta =19.6/W$.

Figure 5

(a) Density of states at different temperatures. (b) The orbitally resolved density of states for each orbital at an inverse temperature $\beta =19.6/W$. (c) The density of states at the Fermi surface of the orbital 3 as a function of inverse temperatures $\beta$. The Coulomb interaction strength is $U/W=0.8$ in all three graphs.

Figure 6

(a) Spectral function for $U/W=0.8$. (b), (c), and (d) are the orbital 1, 2, and 3 parts of the spectral function in (a), respectively. (e) The spectral function for $U/W=2$. (f), (g), and (h) are the orbital 1, 2, and 3 parts of the spectral function in (e), respectively. The dash white line labels the Fermi surface. The inverse temperature is set as $\beta =19.6/W$. Results were obtained with maximum entropy DQMC.

Figure 7

Cartoon sketch of the relevant charge fluctuation processes leading to the insulating state when $U/W=2$ assuming (a) ferromagnetic and (b) antiferromagnetic nearest neighbor correlations within the orbital that has undergone the orbital selective Mott transition (orbital three).

Figure 8

Results for the orbital correlation function for the system in the strong coupling case $U/W=2$. Results are obtained at (a) finite temperature using DQMC and (b) $T=0$ $\left(\beta =\infty \right)$ using DMRG. In both cases, results are shown on $L=8$ (red dots) and $L=16$ (blue triangles) chains. The DQMC results were obtained on a chain with periodic boundary conditions. The DMRG results were obtained on a chain with open boundary conditions.

Figure 9

Results for the orbitally resolved density of states for each orbital obtained for $U/W=2$ and on $L=8$ site chains. Panel (a) shows DQMC results at $\beta =19.6/W$ and the inset zooms in to energy around Fermi surface. Panel (b) shows DMRG results for the same conditions but at zero temperature $\left(\beta =\infty \right)$. The inset plots a finite size scaling analysis of the charge gap obtained within DMRG (see text). The dash line in both panels indicates the Fermi energy.