Ground-state phase diagram of the three-band Hubbard model from density matrix embedding theory

We determine the ground-state phase diagram of the three-band Hubbard model across a range of model parameters using density matrix embedding theory. We study the atomic-scale nature of the antiferromagnetic (AFM) and superconducting (SC) orders, explicitly including the oxygen degrees of freedom. All parametrizations of the model display AFM and SC phases, but the decay of AFM order with doping is too slow compared to the experimental phase diagram, and further, coexistence of AFM and SC orders occurs in all parameter sets. The local magnetic moment localizes entirely at the copper sites. The magnetic phase diagram is particularly sensitive to ${\mathrm{\Delta }}_{pd}$ and $t_{pp}$, and existing estimates of the charge transfer gap ${\mathrm{\Delta }}_{pd}$ appear too large in so-called minimal model parametrizations. The electron-doped side of the phase diagram is qualitatively distinct from the hole-doped side and we find an unusual two-peak structure in the SC in the full model parametrization. Examining the SC order at the atomic scale, within the larger scale $d_{x^2-y^2}$-wave SC pairing order between Cu-Cu and O-O, we also observe a local $p_{x\left(y\right)}$ [or $d_{xz\left(yz\right)}$] symmetry modulation of the pair density on the Cu-O bonds. Our work highlights some of the features that arise in a three-band versus one-band picture, the role of the oxygen degrees of freedom in new kinds of atomic-scale SC orders, and the necessity of re-evaluating current parametrizations of the three-band Hubbard model.

Figure 1

An illustration of the three-band Hubbard model: (a) the symmetric cluster used in the DMET calculations, where the orange and red atoms denote copper and oxygen respectively; (b) definition of the model parameters and the phase convention.

Figure 2

Orbital-projected electronic band structure and density of states (PDOS) of the undoped three-band Hubbard model with Hanke full parameters from HF (left) and DMET (right). The special $\mathbf{k}$ points [$\mathrm{\Gamma }$: $(0,0)$, X: $(\pi ,0)$, M: $(\pi ,\pi )$] are in the first Brillouin zone of the $2\times 2$ supercell lattice. The valence band maximum (VBM) is chosen as the energy zero.

Figure 3

Antiferromagnetic and $d$-wave superconducting order parameters of the hole-doped three-band Hubbard model. The model settings are from the Hanke minimal (left) and Hanke full (right) parameter sets. Note that in the “Hanke full” case, we find two possible solutions between $x=0.2$ and $x=0.4$, marked as solution “1” (from a weakly polarized AFM guess) and “2” (from a strongly polarized AFM guess) in the figure. The curves are cubic-spline interpolated.

Figure 4

Effects of ${\mathrm{\Delta }}_{pd}$ on the magnetic phase diagram of the three-band Hubbard model. ${\mathrm{\Delta }}_{pd}$ ranges from 0.0 to 8.0 eV and the star marker labels the value used in the Hanke model (4.5 eV). Both Hartree-Fock (dotted line) and DMET (shaded area) results are shown.

Figure 5

Effects of $U_d$ on the magnetic phase diagram of the three-band Hubbard model. $U_d$ ranges from 6.0 to 14.0 eV and the star marker labels the value used in the Hanke model (12.0 eV). Both Hartree-Fock (dotted line) and DMET (shaded area) results are shown.

Figure 6

Effects of $U_p$ on the magnetic phase diagram of the three-band Hubbard model. $U_p$ ranges from 0.0 to 8.0 eV and the star marker labels the value used in the Hanke full model (5.25 eV). Both Hartree-Fock (dotted line) and DMET (shaded area) results are shown.

Figure 7

Effects of $t_{pp}$ on the magnetic phase diagram of the three-band Hubbard model. $t_{pp}$ ranges from 0.0 to 2.0 eV and the star marker labels the value used in the Hanke full model (0.75 eV). Both Hartree-Fock (dotted line) and DMET (shaded area) results are shown.

Figure 8

Comparison of electron-doped ($x<0$) and hole-doped ($x>0$) orders. Upper panel: AFM and SC order of the three-band Hubbard model (Hanke full parameter set). Lower panel: AFM and SC order of the one-band Hubbard model ($2\times 2$ DMET cluster), with $U$ fitted such that at $x=0$, $m_{\mathrm{AFM}}$ is the same as that of the three-band model.

Figure 9

Charge, spin, and pairing orders in the three-band Hubbard model. We use yellow and red circles for Cu and O respectively. The area of the circle reflects the corresponding local hole density, the length of the arrow denotes the magnitude of the local magnetic moment, the width of the lines is proportional to the pairing strength, and different colors of the lines denote different coupling signs. The results are calculated based on the fully parametrized model at $x=0.0$ (a), $x=0.3$ doping [solution 1, (b)–(d)] and $x=-0.3$ doping [(e)–(g)]. Panels (b) and (e) show the pairing strength between Cu and Cu; panels (c) and (f) show the pairing strength between the next nearest neighbor O; and panels (d) and (g) illustrate the couplings of both the nearest Cu-O, and the nearest O-O.

Figure 10

DMET energy (in units of $t_{pd}$) and order parameters of the Hybertsen minimal parametrized three-band model, with respect to the number of iterations, at doping $x=0.0$ (left) and $x=0.2$ (right).

Figure 11

DMET energy (in units of $t_{pd}$) and order parameters of the Hybertsen minimal parametrized three-band model, with respect to the discarded weight $\delta$ of the DMRG solver, at doping $x=0.0$ (left) and $x=0.2$ (right). The values are linearly extrapolated to the limit where $\delta =0.0$ (dashed line). The error shown is the standard deviation of linear regression.