Band crossover and magnetic phase diagram of the high-$T_c$ superconducting compound ${\mathrm{Ba}}_2{\mathrm{CuO}}_{4-\delta }$

We present the influences of electronic and magnetic correlations and doping on the ground-state properties of the recently discovered superconductor ${\mathrm{Ba}}_2{\mathrm{CuO}}_{4-\delta }$ by utilizing the rotationally invariant slave-boson method. Starting with an effective two-orbital Hubbard model [Scalapino et al., Phys. Rev. B 99, 224515 (2019)], we demonstrate that with increasing doping concentration, the paramagnetic (PM) system evolves from a two-band character to single band around the electron filling $n=2.5$, with the band nature of the $d_{3z^2-r^2}$ and $d_{x^2-y^2}$ orbitals to the $d_{3z^2-r^2}$ orbital, slightly affected when the electronic correlation $U$ varies from 2 to 4 eV. Considering the magnetic correlations, the system displays one antiferromagnetic (AFM) metallic phase in $2<n<2.14$ and a PM phase in $n>2.14$ at $U=2\mathrm{eV}$, or two AFM metallic phases in $2<n<2.52$ and $2.74<n<3$, and a PM phase in $2.52<n<2.74$, respectively, at $U=4\mathrm{eV}$. Our results show that near the realistic superconducting state around $n=2.6$ the intermediate correlated ${\mathrm{Ba}}_2{\mathrm{CuO}}_{3,2}$ should be of single-band character. The $s$-wave superconducting pairing strength becomes significant when $U>2\mathrm{eV}$, and crosses over to $d$ wave when $U>2.2\mathrm{eV}$.

Figure 1

The 16 local Fock states for the two-orbital Hubbard model.

Figure 2

Two possible spin configurations of the Cu spins of (a) Néel antiferromagnet and (b) collinear antiferromagnet. The gray balls represent Cu atoms, and the red and blue arrows denote the spin directions.

Figure 3

The dependence of the band structures on different doping in PM ${\mathrm{Ba}}_2{\mathrm{CuO}}_{4-\delta }$ at $U=0\mathrm{eV}$ (blue line), $U=2\mathrm{eV}$ (red line), and $U=4\mathrm{eV}$ (black line).

Figure 4

The Fermi surfaces of ${\mathrm{Ba}}_2{\mathrm{CuO}}_{4-\delta }$ for different doping at $U=0\mathrm{eV}$ (blue line), $U=2\mathrm{eV}$ (red line), and $U=4\mathrm{eV}$ (black line).

Figure 5

The $U$ dependence of the total energy $E_{\mathrm{tot}}$ of three different magnetic phases (a) and of the total, band, and interaction energy differences $\mathrm{\Delta }{E}_{\mathrm{tot}}, \mathrm{\Delta }{E}_{\mathrm{band}}$, and $\mathrm{\Delta }{E}_{\mathrm{int}}$ between the Néel AFM and PM phases (b). Here PM, AFM metallic, and AFI denote the paramagnetic metallic phase, Néel antiferromagnetic metallic, and antiferromagnetic insulating phases at $n=2$, respectively.

Figure 6

The $U$ dependence of total energies of ${\mathrm{Ba}}_2{\mathrm{CuO}}_{4-\delta }$ in the PM and Néel AFM configurations and of the energy differences $\mathrm{\Delta }{E}_{\mathrm{tot}},\mathrm{\Delta }{E}_{\mathrm{band}},\mathrm{\Delta }{E}_{\mathrm{int}}$ at $n=3$.

Figure 7

The magnetic phase diagrams of ${\mathrm{Ba}}_2{\mathrm{CuO}}_{4-\delta }$ and the energy differences between the PM state and the Néel AFM state at $U=2\mathrm{eV}$ (a) and $U=4\mathrm{eV}$ (b).

Figure 8

The sublattice magnetic moments of ${\mathrm{Ba}}_2{\mathrm{CuO}}_{4-\delta }$ at $U=2\mathrm{eV}$ (dashed line) and $U=4\mathrm{eV}$ (solid line) as the functions of band filling.

Figure 9

The evolutions of the Fermi surfaces of ${\mathrm{Ba}}_2{\mathrm{CuO}}_{4-\delta }$ with increasing doping at $U=2\mathrm{eV}$ (upper four panels) and $U=4\mathrm{eV}$ (lower four panels). The color represents the orbital weight of the $d_{x^2-y^2}$ (0) and $d_{3z^2-r^2}$ (1) orbitals.

Figure 10

The evolutions of the superconducting pairing strength $\lambda$ of ${\mathrm{Ba}}_2{\mathrm{CuO}}_{4-\delta }$ in the $d$-wave channel with the increasing correlations at $n=2.6$. Inset shows the dependence of the superconducting pairing strength $\lambda$ on increasing doping at $U=2\mathrm{eV}$.