Spatial Correlations and the Insulating Phase of the High-$T_c$ Cuprates: Insights from a Configuration-Interaction-Based Solver for Dynamical Mean Field Theory

A recently proposed configuration-interaction-based impurity solver is used in combination with the single-site and four-site cluster dynamical mean field approximations to investigate the three-band copper oxide model believed to describe the electronic structure of high transition temperature copper-oxide superconductors. Use of the configuration interaction solver enables verification of the convergence of results with respect to the number of bath orbitals. The spatial correlations included in the cluster approximation substantially shift the metal-insulator phase boundary relative to the prediction of the single-site approximation and increase the predicted energy gap of the insulating phase by about 1 eV above the single-site result. Vertex corrections occurring in the four-site approximation act to dramatically increase the value of the optical conductivity near the gap edge, resulting in better agreement with the data. The calculations reveal two distinct correlated insulating states: the “magnetically correlated insulator,” in which nontrivial intersite correlations play an essential role in stabilizing the insulating state, and the strongly correlated insulator, in which local physics suffices. Comparison of the calculations to the data places the cuprates in the magnetically correlated Mott insulator regime.

Figure 1

Density of states from (a) four-site and (b) single-site DMFT with various values of $N_b$. A small broadening factor $\eta =0.10\mathrm{eV}$ is used.

Figure 2

Metal-insulator transition phase diagram in the plane of interaction strength $U$ and $p\text{−}d$ energy splitting $\mathrm{\Delta }$ (a) and $d$-occupancy $N_d$ (b) from single-site ($N_c=1$) and four-site ($N_c=4$) dynamical mean field approximation. The error bars reflect uncertainties arising from restricting to $P=2$ particle-hole pairs at large $U$; where not shown, they are smaller than the size of the points. The region shaded with lines slanting up and to the left is the conventional Mott insulator (CMI) region, where insulating behavior is found even for $N_c=1$; the region shaded with lines slanting up and to the right indicates regions that are metallic in the single-site approximation, and the cross-hatched region is the magnetically correlated Mott insulator (MCMI), which is insulating for $N_c=4$ but not $N_c=1$. The shaded area is the coexistence region of the metal-insulator transition in the single-site approximation. The cross and the star denote parameter values that yield gaps comparable to the experimental values for $N_c=1$ and 4, respectively.

Figure 3

Spectral gap as a function of $d$ occupancy for various values of $U$ with $(N_c,N_b)=(1,9)$ and (4,12). For $N_c=1$, two distinct solutions depending on initial conditions are shown in the intermediate region. The black horizontal line indicates the charge transfer gap ${\mathrm{\Delta }}_{\rho }=2\mathrm{eV}$ characteristic of the parent compounds of the high-$T_c$ cuprates. The size of the shaded region reflects the uncertainties arising from the $P=2$ approximation at large $U$.

Figure 4

(a) Optical conductivity from the single-and four-site DMFT with parameters fixed to yield gap size ${\mathrm{\Delta }}_{\rho }\sim 2\mathrm{eV}$ at correlation strength $U=8.0\mathrm{eV}$, while the four-site calculation leads to a rapid rise in the conductivity at the gap edge, in agreement with experiment. (b) Vertex and bubble contribution to the optical conductivity for four-site DMFT. (c) Density of states with the same parameters in (a).