Metal-insulator transition in the ground state of the three-band Hubbard model at half filling

The three-band Hubbard model is a fundamental model for understanding properties of the copper-oxygen planes in cuprate superconductors. We use cutting-edge auxiliary-field quantum Monte Carlo (AFQMC) methods to investigate ground state properties of the model in the parent compound. Large supercells combined with twist averaged boundary conditions are employed to reliably reach the thermodynamic limit. Benchmark quality results are obtained on the magnetic correlations and charge gap. A key parameter of this model is the charge-transfer energy $\mathrm{\Delta }$ between the oxygen $p$ and the copper $d$ orbitals, which appears to vary significantly across different families of cuprates and whose ab initio determination is subtle. We show that the system undergoes a quantum phase transition from an antiferromagnetic insulator to a paramagnetic metal as $\mathrm{\Delta }$ is lowered to $3\mathrm{eV}$.

Figure 1

(Left) Schematic view of the ${\mathrm{CuO}}_2$ plane of the cuprates. Cu $3d_{x^2-y^2}$ orbitals are represented in blue, and $O2p_x$ and $2p_y$ orbitals in green. The curve connectors represent the hopping, and the labels define the sign rule. (Right) Density of holes around the $d$ and the $p$ sites ($n_d+2n_p=1$) as a function of $\mathrm{\Delta }$. Results computed from AFQMC are given by blue circles. The green boxes indicate the typical values observed (or extrapolated from) in families of cuprate materials [19, 20]: $n_d\simeq 0.83$ for La-based, $\simeq 0.7$ for Y-based, and $\simeq 0.5$ for Bi- and Hg-based cuprates.

Figure 2

Computed ground-state magnetic properties. The upper panel shows a color plot of the spin correlation function at $\mathrm{\Delta }=4.4$ eV for a $12\times 12$ supercell. The lower panel shows the order parameter at asymptotic distances for a sequence of $\mathrm{\Delta }$ values.

Figure 3

Metal-insulator transition as a function of the charge transfer energy $\mathrm{\Delta }$. Three different signatures are computed: antiferromagnetic order parameter $\left|S\right(\vec{r}\left)\right|$ (upper panel), charge gap ${\mathrm{\Delta }}_C$ as defined in Eq. (4) (middle panel), logarithm of the localization measure in Eq. (3) (lower panel). The values of $-ln\left|\zeta \right|$ for $\mathrm{\Delta }<3$ eV are compatible with $+\infty$. The shaded area indicates the phase transition region.

Figure 4

(Left panel) Distance dependence of the $d$-wave pairing correlation function $C_{\mathrm{\Delta }}\left(\mathbf{r}\right)$ for a few values of the charge transfer energy. (Right panel) Density correlation function $C_C\left(\mathbf{r}\right)$ plotted along the Cu-O bond. We denote $x$ the coordinate along the bond, as in Fig. 1. We use open symbols for correlations involving one $d$ orbital and one $p_x$, while solid symbols indicate $d-d$ correlations. The noninteracting result, which does not depend on $\mathrm{\Delta }$, is also shown (black dotted line) for reference. The inset is a zoom in, for $x>0.5$.