Doping-driven metal-insulator transition in correlated electron systems with strong Hund's exchange coupling

We study the doping-driven Mott metal-insulator transition for multiorbital Hubbard models with Hund's exchange coupling at finite temperatures. As in the single-orbital Hubbard model, the transition is first order within dynamical mean-field theory, with a coexistence region where two solutions can be stabilized. We find that in the presence of finite Hund's coupling, the insulating phase is connected to a badly metallic phase, which extends to surprisingly large dopings. While fractional power-law behavior of the self-energies on the Matsubara axis is found on both sides of the transition, a regime with frozen local moments develops only on the branch connected to the insulating phase.

Figure 1

Top: Phase diagram for the two-orbital model with density-density interaction at constant $J=0.25U$. The coexistence region constitutes a volume in the $(T,U,\mu )$ parameter space and is indicated by the blue cuts. Black dots mark the positions of the spinoidals (defining the boundary of the coexistence region), evaluated from the calculations performed in this work. The dashed lines ${\mu }_{C1}, {\mu }_{C2}$, and $T_C$ are guides to the eye, indicating the critical chemical potentials in a plane of constant temperature ($T=0.005$, i.e., $\beta =200$, the lowest temperature we have simulated here) and the critical temperature. Bottom: Evaluated coexistence region for $J=0.25U$ (black dots) in the $(U,T)$ plane, compared to the case $J=0$ (schematic taken from Ref. [49]).

Figure 2

Filling $n$ per spin as a function of the chemical potential $\mu$ for the two-orbital model (3) with $J=0.25U, \beta =100$ (top) and without Hund's coupling $J=0, \beta =25$ (bottom) for various Hubbard interactions $U$. Solid (dashed) lines denote results starting from a metallic (insulating) initial configuration.

Figure 3

Imaginary part of the self-energies (top) and Green's functions (bottom) for the two-orbital model equation (3) with $U=3.06, J=0.25U$, and $\beta =100$. Solid (dashed) lines denote simulations starting from a metallic (insulating) initial configuration.

Figure 4

Top: Filling $n$ per spin as a function of $\mu$ in the two-orbital model (3) for different temperatures $T=1/\beta$, with $U=3.06$ and $J=0.25U$. The curve for $\beta =25$ has been fitted to $n=A{(\mu -{\mu }_0)}^2$, yielding $A=0.27, {\mu }_0=0.19$. Bottom: Filling $n$ per spin as a function of $\mu$ in the two-orbital model (3) for different temperatures $T=1/\beta$, with $U=7.35$ and $J=0$. Solid (dashed) lines designate solutions starting from a metallic (insulating) configuration.

Figure 5

Self-energies of the two-orbital model for several chemical potentials $\mu$, with $U=3.06, J=0.25U$, and inverse temperature $\beta =200$. Dashed lines indicate least-squares fits to the function $c+{b\left|i\omega \right|}^{\delta }$. Left: Imaginary part of the self-energy on the Matsubara axis on the insulating branch of the coexistence region, $\delta =0.72,0.75,0.66,0.50$ (for $\mu =0.30,0.34,0.38,0.42$). Right: Same quantity on the metallic branch, $\delta =0.45,0.46,0.48,0.50$ (for $\mu =0.30,0.34,0.38,0.42$).

Figure 6

Real and imaginary parts of the self-energies on the Matsubara axis for the two-orbital model with $U=3.06, J=0.25U$, and $\beta =200$ for constant doping $n=0.515$/spin on the metallic (blue) and insulating (red) branches.

Figure 7

Real and imaginary parts of the self-energies on the real-frequency axis for the two-orbital model with $U=3.06, J=0.25U$, and $\beta =200$ for doping $n=0.515$/spin on the metallic (blue) and insulating (red) branches. The inset shows a zoom of the metallic branch around the Fermi level.

Figure 8

Spectral functions on the metallic and insulating branches for fixed filling $n=0.515$/spin, $U=3.06, J=0.25$, and $\beta =200$.

Figure 9

Spectral functions $A(\omega ,\epsilon )=-\text{Im}\left[G\right(\omega ,\epsilon \left)\right]/\pi$ on the metallic and insulating branches for fixed filling $n=0.515$/spin, $U=3.06, J=0.25$, and $\beta =200$. The Green's functions were calculated as $G(\omega ,\epsilon )={[\omega +\mu -\epsilon -\mathrm{\Sigma }\left(\omega \right)]}^{-1}$ after analytic continuation of the self-energies $\mathrm{\Sigma }\left(i\omega \right)$.

Figure 10

Spin-spin correlations $C_{ss}\left(\tau \right)=〈S_z\left(\tau \right)S_z\left(0\right)〉$ for the two-band model at $U=3.06$, with $J=0.25U$. Left: The solid (dashed) lines show the spin-spin correlation functions at fillings $n=0.53$ ($n=0.51$) for different temperatures. Right: Blue dots (red triangles) mark the correlations at $\tau =\beta /2$ for $n=0.53$ ($n=0.51$) as a function of the temperature $T={\beta }^{-1}$. The dashed line is a linear fit (for $n=0.53$).

Figure 11

Filling $n$ per spin as a function of the chemical potential $\mu$ for the two-orbital model with spin-flip and pair-hopping terms ($\alpha =1$) for various temperatures, $U=3.4$, and $J=0.25U$. Solid (dashed) lines denote results starting from a metallic (insulating) initial configuration.

Figure 12

Imaginary part of the self-energies for the two-orbital model with spin-flip and pair-hopping terms in the interaction Hamiltonian, i.e., $\alpha =1$. $U=3.4, J=0.25U$, and $\beta =100$; solid (dashed) lines denote results starting from a metallic (insulating) initial configuration.

Figure 13

Filling $n$ per spin as a function of the chemical potential $\mu$ for a three-band model with $U=2.3, J=0.25U$, and $\beta =50$. Solid (dashed) lines denote results starting from a metallic (insulating) initial configuration. Inset: $n\left(\mu \right)$ per spin for $U=2.25,2.3,2.6$ (blue, black, and green, respectively) for $J=0.25$, starting from a metallic configuration.