Correlation effects and concomitant two-orbital $s_{\pm }$-wave superconductivity in ${\mathrm{La}}_3{\mathrm{Ni}}_2{\mathrm{O}}_7$ under high pressure

High-$T_c$ superconductivity (SC) has been found experimentally in the bilayer material ${\mathrm{La}}_3{\mathrm{Ni}}_2{\mathrm{O}}_7$ under high pressure recently, in which the Ni-$3d_{3z^2-r^2}$ and $3d_{x^2-y^2}$ orbitals are expected to play a key role in the electronic structure and the SC. Here we study the two-orbital electron correlations and the nature of the SC in the framework of the dynamical mean-field theory using the bilayer two-orbital Hubbard model downfolded from the band structure of ${\mathrm{La}}_3{\mathrm{Ni}}_2{\mathrm{O}}_7$. We find that each of the two orbitals forms $s_{\pm }$-wave SC pairing. Because of the interorbital hoppings, the two-orbital SCs are concomitant, and furthermore they transition to Mott insulating states simultaneously when tuning the system to half filling. The Hund's coupling induced local interorbital spin coupling enhances the electron correlations pronouncedly and is crucial to the SC.

Figure 1

Electron density of a Ni atom as a function of the chemical potential $\mu$. (a) Electron densities for different orbitals. Inset: enlarged view around half-filling. When $\mu >-2$, the increase of $n_z$ becomes slow. When $\mu$ exceeds $-0.7$, $n_z$ surpasses one without the $z$ orbitals being Mott insulating. As $n_x$ gradually approaches one, $n_z$ drops back to one, and the two orbitals simultaneously transition from SC to Mott insulating at $\mu =0$. (b) Electron densities for the bonding and antibonding states of different orbitals. As $\mu$ decreases from zero, $n_{z+}$ increases counterintuitively and $n_{z-}$ is not small, implying that the $z$ orbitals are strongly correlated. A negative electronic compressibility is observed on the bonding state of the $z$ orbital.

Figure 2

SC order parameters vs the chemical potential $\mu$. (a) Local (${\mathrm{\Delta }}_{0\alpha }$) and interlayer (${\mathrm{\Delta }}_{1\alpha }$) pairings. In the whole SC regime, the interlayer pairing is dominant, and ${\mathrm{\Delta }}_{1x}$ is larger than ${\mathrm{\Delta }}_{1z}$. (b) SC order parameters for the bonding (${\mathrm{\Delta }}_{\alpha +}$) and antibonding (${\mathrm{\Delta }}_{\alpha -}$) states. Both of the two orbitals are $s_{\pm }$-wave superconducting and become Mott insulating simultaneously at $\mu =0$.

Figure 3

(a) Electron densities for different orbitals and (b) SC order parameters as functions of chemical potential $\mu$ after $t_3^{xz}$ and $t_4^{xz}$ being suppressed to zero. Solid lines represent order parameters for normal hopping parameters, and dashed lines for hopping parameters with $t_3^{xz}$ and $t_4^{xz}$ suppressed to zero. An orbital selective Mott (or SC) phase has been observed after suppressing the interorbital hopping parameters $t^{xz}$. All SC order parameters have been suppressed except ${\mathrm{\Delta }}_{1x}$.

Figure 4

(a) Electron densities and (b) SC order parameters with $\mathrm{\Delta }{U}^{\prime }=U_{\mathrm{a}}^{\prime }-U_{\mathrm{p}}^{\prime }=0.002$ and $U_{\mathrm{a}}^{\prime }+U_{\mathrm{p}}^{\prime }$ unchanged. An almost linear relation for both $n_x$ and $n_z$ as $\mu$ increases. In (b), solid lines represent order parameters for $\mathrm{\Delta }{U}^{\prime }=1$ and dashed lines for $\mathrm{\Delta }{U}^{\prime }=0.002$. ${\mathrm{\Delta }}_{1x}$ has been suppressed to a small magnitude, and other SC order parameters almost disappear.

Figure 5

SC order parameters vs electron density for different values of hopping parameters. Solid lines represent order parameters for normal hopping parameters, and dashed lines for hopping parameters which are suppressed. (a) $t_{\perp }^{\alpha }$ for both orbitals being reduced by 10%. (b) $t_3^{xz}$ and $t_4^{xz}$ being reduced by 10%.