Superconductivity of bilayer two-orbital Hubbard model for La$_{3}$Ni$_{2}$O$_{7}$ under high pressure

Abstract

By combining density functional theory (DFT) and density matrix renormalization group calculations, we investigate the unusual pressure dependence of superconducting transition temperature ($T_c$) in the nickelate superconductor La$_{3}$Ni$_{2}$O$_{7}$. Using the hopping integrals and on-site potentials obtained by fitting the DFT band structures, we map a quantum phase diagram of a bilayer two-orbital Hubbard model with increasing pressure in a ladder geometry, which has an intermediate Hubbard repulsion and a Hund's coupling. Near $3/8$ filling, we find a strong spin density wave order, which at $3/8$ filling shows a real-space spin pattern similar to the spin-charge stripe order along a lattice direction. At $21/64$ filling, we find a superconducting phase with interlayer superconductivity (SC) in both the $d_{z^2}$ and $d_{x^2-y^2}$ orbitals, as well as in-plane SC in the $d_{z^2}$ orbital. Intriguingly, the SC is weakened with increasing pressure and transits to a Luttinger liquid above $80$ GPa, which qualitatively agrees with the experimental observations of decreasing $T_c$ with increasing pressure and a transition to Fermi liquid above $80$ GPa in La$_{3}$Ni$_{2}$O$_{7}$. Through a comparative study, we further show that the ratio of interaction to hopping integral, which reduces moderately with increasing pressure, may play a dominant role in the weakening of SC. Our results of this experimentally relevant model not only find a robust SC through suppressing the competing spin density wave order, but also give new insight into the unusual pressure dependence of SC in La$_{3}$Ni$_{2}$O$_{7}$.

Figures (14)

• Figure 1: (a) Crystal structures of La$_3$Ni$_2$O$_7$. The middle and right panels denote the primitive cells of $Fmmm$ and $I4/mmm$ phases. (b) DFT calculated band structures of primitive cell La$_3$Ni$_2$O$_7$ under various pressures. The bottom panel shows a zoom in of the flat bonding band ($\gamma$) near the Fermi level at the $M$ corner of the Brillouin zone. (c) Comparison of the DFT band structure (gray) and the fitted band structure from TB model (colored) under the pressure of $60.3$ GPa. (d) Fermi surface with one hole pocket ($\gamma$) and two electron pockets ($\alpha$ and $\beta$) of the TB model at $60.3$ GPa. The color bar denotes the orbital weight of the $d_{x^2-y^2}$ and $d_{z^2}$ orbitals.
• Figure 2: (a) Schematic figure of the bilayer two-orbital model with the $d_{z^2}$ and $d_{x^2-y^2}$ orbitals. The hopping integrals $t_{||}^{zz}$, $t_{||}^{xz}$, $t_{||}^{xx}$, and $t_{\perp}^{zz}$ are chosen from the TB model shown in Table \ref{['tab1']}. (b) DMRG phase diagram of the model on a ladder geometry with system width $L_y=1$ and the interactions $U = 4.0$ eV and $J_H = 0.5$ eV. By tuning electron filling per unit cell $\eta_e$ and pressure, the system exhibits a spin density wave (SDW) phase (yellow circle), a superconducting phase (red star), and a Luttinger-liquid (LL) phase (purple pentagon). At $\eta_e = 21/64$, the system has a transition from SC to LL near $80$ GPa (red star in a circle).
• Figure 3: Electron density distributions ($n_z$ and $n_x$) and density correlation functions ($D_z$ and $D_x$) of both $d_{z^{2}}$ (triangle) and $d_{x^{2}-y^{2}}$ (circle) orbitals for the electron filling $\eta_e = 3/8$ and $21/64$ under different pressures. The insets in (c) and (g) show the Fourier transform of electron density distribution in the $d_{x^{2}-y^{2}}$ orbital. $K^{\mu}_D$ ($\mu = z$ or $x$) denotes the obtained power exponents by the algebraic fitting of density correlation functions.
• Figure 4: Local magnetic moments ($m_{z}$ and $m_x$) and spin correlation functions ($F^{zz}_{z}$ and $F^{zz}_x$) of both $d_{z^{2}}$ (triangle) and $d_{x^{2}-y^{2}}$ (circle) orbitals for the electron filling $\eta_e = 3/8$ and $21/64$ under different pressures. Edge pinning magnetic field is introduced to compute local magnetic moments. Spin correlation functions for $\eta_e = 3/8$ are plotted as double-logarithmic scale with the fitted power exponents $K^{\mu}_s$. For $\eta_e = 21/64$, spin correlations are presented as semi-logarithmic scale with the spin correlation lengths $\xi^{\mu}_s$. The insets in (b) and (d) show the corresponding spin structure factor at $\eta_e = 3/8$.
• Figure 5: (a) Comparisons of the interlayer pairing correlation function (solid) and the square of single-particle Green's function (dotted) in both $d_{x^{2}-y^{2}}$ (circular) and $d_{z^{2}}$ (triangle) orbitals under different pressures for $\eta_e = 21/64$. The interlayer pairing correlation functions $P^{\perp}_x(r)$ and $P^{\perp}_{z}(r)$ are fitted algebraically, giving the power exponents $K^x_{SC}$ and $K^{z}_{SC}$, respectively. (b) Comparison of SC transition temperature $T_c$ by the high-pressure resistance measurements 10.1093/nsr/nwaf220 and the power exponents $-K^{\mu}_{SC}$ obtained in subfigure (a).