On the nature of the Mott transition in multiorbital systems

We analyze the nature of a Mott metal-insulator transition in multiorbital systems using dynamical mean-field theory. The auxiliary multiorbital quantum impurity problem is solved using continuous-time quantum Monte Carlo and the rotationally invariant slave-boson (RISB) mean-field approximation. We focus our analysis on the Kanamori Hamiltonian and find that there are two markedly different regimes determined by the nature of the lowest-energy excitations of the atomic Hamiltonian. The RISB results at $T\to 0$ suggest the following rule of thumb for the order of the transition at zero temperature: a second-order transition is to be expected if the lowest-lying excitations of the atomic Hamiltonian are charge excitations, while the transition tends to be first order if the lowest-lying excitations are in the same charge sector as the atomic ground state. At finite temperatures, the transition is first order and its strength, as measured, e.g., by the jump in the quasiparticle weight at the transition, is stronger in the parameter regime where the RISB method predicts a first-order transition at zero temperature. Interestingly, these results seem to apply to a wide variety of models and parameter regimes.

Figure 1

Lowest-lying multiplet structure of Hamiltonian (6) at half filling and $\delta \ll U$. For $U^{\prime }/U>1/2$, the lowest-lying states in the charge sectors with $N=M-1$ and $N=M+1$ particles are lower in energy than the states in the charge sector with $N=M$ and one or more orbitals having double occupancy.

Figure 2

(a) Quasiparticle weight $Z$ vs $U$ obtained using RISB for different values of $U^{\prime }/U=0,0.2,0.5,0.6,0.7,0.8,0.9,0.98,0.99,$ and 1. The ratio $U^{\prime }/U$ increases from left to right as indicated in the figure. Blue disks are used for $U^{\prime }/U\le 0.5$, black disks for $0.5<U^{\prime }/U<1$, and red disks for $U^{\prime }/U=1$. Inset: Zoom of $U_{c2}$ for $U^{\prime }/U\le 0.8$. (b) Free-energy difference between the metallic and insulating solutions for the parameters of the upper panel. Inset: $U$ – $U^{\prime }$ phase diagram.

Figure 3

(a) Derivative of the difference between the free energies of the metal and insulator solutions at the free-energy crossing point within the RISB method as function of $U^{\prime }/U$. (b) Same data scaled with the single-orbital temperature dependence. (c) Quasiparticle weight at $U_c$ scaled by its single-orbital temperature dependence.

Figure 4

Total statistical weight of local multiplets in the partition function calculated with the RISB method for three values of the ratio $U^{\prime }/U$ (1, 0.9, and 0.2). For $U^{\prime }/U=1$, the six $2PS$ and $2PD$ states are degenerate and have the same weight. In the insulating phase, within the RISB approximation, there are no charge fluctuations and the full statistical weight is on the ground-state sector of the atomic Hamiltonian.

Figure 5

Quasiparticle weight $Z$ vs $U$ obtained using RISB (top) and CTQMC (bottom) techniques as impurity solver for $\beta D=200$.

Figure 6

Statistical weight of local multiplets in the partition function as a function of $U/U_{2c}$ for the two-orbital case and $J=0$. The results were obtained at $\beta D=200$ using RISB (lines) and CTQMC (symbols) as impurity solvers. The insets present in each case the CTQMC weights close to the transition.

Figure 7

Quasiparticle weight for the half-filled three-orbital model given by Eq. (6) and for different values of $U^{\prime }/U=0.0,0.2,0.4,0.5,0.6,0.7,0.8,0.9,0.98,0.99,$ and 1. Inset: quasiparticle weight close to the transition at half filling for two and three orbitals and $U^{\prime }/U=0.9$.

Figure 8

(a) Quasiparticle weight $Z$ vs $U/D$ for different values of $J/U$ in a half-filled three-orbital Kanamori system. (b) Statistical weight for high-spin $\left(S=3/2\right)$ and low-spin $\left(S=1/2\right)$ states. The inset presents the structure of the lowest-lying excitations of the atomic Hamiltonian for $J/U=0.25$ and $J/U=0.05$. $\mathrm{\Delta }E$ is the energy difference between the lowest-lying excited states and the ground state, which is in the $N=3$ charge sector and has a spin $S=3/2$.

Figure 9

(a) Quasiparticle weight $Z$ vs $U/D$ for different values of $J/U$ in a three-orbital Kanamori system with an occupation of two electrons per site. (b) Statistical weight for high-spin $\left(S=1\right)$ and low-spin $\left(S=0\right)$ states. The inset shows the structure of the lowest-lying excitations of the atomic Hamiltonian for $J/U=0.15$ and $J/U=0.01$. $\mathrm{\Delta }E$ is the energy difference between the lowest-lying excited states and the ground state which is in the $N=2$ charge sector and has a spin $S=1$.