Orbital-selective Mott transition in multiband systems: Slave-spin representation and dynamical mean-field theory

We examine whether the Mott transition of a half-filled, two-orbital Hubbard model with unequal bandwidths occurs simultaneously for both bands or whether it is a two-stage process in which the orbital with narrower bandwith localizes first (giving rise to an intermediate “orbital-selective” Mott phase). This question is addressed using both dynamical mean-field theory and a representation of fermion operators in terms of slave quantum spins, followed by a mean-field approximation (similar in spirit to a Gutzwiller approximation). In the latter approach, the Mott transition is found to be orbital selective for all values of the Coulomb exchange (Hund) coupling $J$ when the bandwidth ratio is small and only beyond a critical value of $J$ when the bandwidth ratio is larger. Dynamical mean-field theory partially confirms these findings, but the intermediate phase at $J=0$ is found to differ from a conventional Mott insulator, with spectral weight extending down to arbitrary low energy. Finally, the orbital-selective Mott phase is found, at zero temperature, to be unstable with respect to an interorbital hybridization $V$ and replaced at small $V$ by a state with a large effective mass (and a low quasiparticle coherence scale) for the narrower band.

Figure 1

(Color online) Filling vs chemical potential for a two-orbital impurity (atomic limit of a particle-hole-symmetric Hubbard model), $U=2$, $\beta =50$: within the slave-spin mean-field (solid line) and exact (dashed line) results. The Coulomb staircase is correctly reproduced up to temperatures of order $\sim U$.

Figure 2

(Color online) Quasiparticle weight, obtained from slave-spin mean-field theory, for the $N$-orbital Hubbard model at half-filling (with, from left to right, $N=1,2,3,4$). The noninteracting density of states is a semicircle with half-bandwidth $D$. Inset: Dependence of the critical $U$ on $N$. The exact large-$N$ behavior is obtained.

Figure 3

(Color online) Quasiparticle weight of the one-band Hubbard model, obtained with slave rotors (thin line), slave spins, and the Gutzwiller (slave-boson) approximations (thick lines). The latter two actually coincide. The small-$U$ behavior of the slave-rotor approach is due to the larger number of unphysical states (see text).

Figure 4

(Color online) Phase diagram ($U$ vs $U^{\prime }$) for $t_2∕t_1=0.5$ at $T=0$ within the slave-spin mean-field theory. Inset: same diagram obtained with exact diagonalization dynamical mean-field theory (ED-DMFT) in Ref. 9. The dotted line indicates $J=0$—i.e., $U=U^{\prime }$. In “phase I” both bands are metallic; in “phase II” both bands are insulating. “Phase III” is the orbital-selective Mott phase.

Figure 5

Dependence of the critical $U$ on the ratio $t_2∕t_1$ at $J=0$ and $T=0$. All transitions are second order.

Figure 6

$Z_m$ at $J=0$ and $T=0$ for $t_2∕t_1=0.5$ (top), 0.25 (middle), 0.15 (bottom).

Figure 7

(Color online) Widening of the OSMT zone with increasing $J∕U$ at $T=0$. Transition lines are shown for (from left to right) $J∕U=0$ (dashed line), 0.01, 0.1.

Figure 8

(Color online) Dependence of the critical $J∕U$ above which the Mott transition becomes orbital selective on the ratio $t_2∕t_1$ at $T=0$.

Figure 9

(Color online) Phase diagram ($U$ vs $U^{\prime }$) for $t_2∕t_1=0.5$ at $T=0$ for a two-band Hubbard model without spin-flip and pair-hopping terms in the interaction.

Figure 10

(Color online) Critical $J∕U$ for the model without spin-flip and pair-hopping terms as a function of $t_2∕t_1$. At small ratios of the bandwidths, the system displays an OSMT above a critical $J∕U$ ratio which vanishes at $t_2∕t_1=0.2$. In less anisotropic systems large Hund’s couplings are needed to realize OSMP’s, whereas beyond the critical bandwidth ratio of 0.6 no OMST is possible within this model.

Figure 11

(Color online) Same phase diagram as in Fig. 9 but in the $U\text{−}J$ plane at $T=0$ and at $\beta t_1=40$.

Figure 12

(Color online) Quasiparticle residues at $J=0$ in DMFT (ED) for $t_2∕t_1=0.1$ (symbols). For comparison, the slave-spin mean-field approximation is also displayed (solid line). Inset: same quantities as a function of $t_2∕t_1$ at fixed $U∕t_1=4.0$. (The kinks in $Z_1$ are associated with the criticality of $Z_2$.)

Figure 13

(Color online) Phase diagram at $J=0$ and $T=0$ in ED DMFT. The dashed line marks the disappearance of the Mott gap (downward runs). For $U$ values around 5 there is coexistence between an OSMP and—depending on the bandwidth ratio—an insulating or metallic phase.

Figure 14

Imaginary part of the Green functions of the two bands at low energy for $t_2∕t_1=0.16$. Upper panel: $U∕t_1=7.0$, both bands are insulating. Central panel: $U∕t_1=4.0$, orbital-selective Mott phase. Lower panel: $U∕t_1=2.0$, both bands metallic. The dashed curves correspond to the orbital with narrower bandwidth. All energy scales are in units of $t_1$.

Figure 15

(Color online) Imaginary parts of the Green’s functions in Matsubara space for $t_2∕t_1=0.1$, $U∕t_1=1.6$. Solid and dashed lines represent the exact diagonalization results for the wide and narrow bands, respectively. Circles and squares represent QMC data for the same quantities at $\beta t_1=40$.

Figure 16

(Color online) Spectral function for $t_2∕t_1=0.1$, $U∕t_1=1.6$ at $\beta t_1=40$ in QMC DMFT. The solid and dashed red lines denote the narrow and wide bands, respectively. The DOS of the noninteracting system (wide band only) is given for comparison.

Figure 17

(Color online) Quasiparticle residues $Z_1$ and $Z_2$ within slave-spin MFT for finite $V$. Top: $t_2∕t_1=0.5$, $J=0.25U$ for (from right to left in the upper manifold, wide band; from left to right in the lower manifold, narrow band): $V=0$ (OSMT system), $V∕t_1=0.1,0.15,0.2,0.3$. Bottom: $t_2∕t_1=0.1$, $J=0$ for (from right to left in the upper manifold, wide band; from left to right in the lower manifold, narrow band) $V=0$ (OSMT system), $V∕t_1=0.05,0.1,0.2$. Insets show the same graph in logarthmic scale.

Figure 18

(Color online) Gap amplitude of the auxiliary fermions (see text) and quasiparticle residues within slave-spin MFT for two values of $V$. Top: $t_2∕t_1=0.1$, $J=0$ for $V∕t_1=0.1$ (upper panel) and $V∕t_1=0.6$ (lower panel). The corresponding critical $V$ for the noninteracting system is $\sqrt{D_1{D}_2}\simeq 0.63$ in this case.