Ground state of doped cuprates from first-principles quantum Monte Carlo calculations

The author reports on high-fidelity simulations of charge carriers in the high-$T_c$ cuprate materials using quantum Monte Carlo techniques applied to the first-principles Hamiltonian. With this high accuracy technique, the doped ground state is found to be a spin polaron, in which charge is localized through a strong interaction with the spin. This spin polaron has calculated properties largely similar to the phenomenology of the cuprates, and may be the object which forms the Fermi surface and charge inhomogeneity in these materials. The spin polaron has some unique features that should be visible in x-ray, EELS, and neutron experiments.

Figure 1

Assessing errors in the calculations for a $2\times 2\times 1$ supercell, with $x=0.25$. (a) The dependence of the relative magnetic energies versus the density functional used to generate the nodes. All structures have a minimum within stochastic uncertainties at around 25% mixing. (b) At 25% mixing, the dependence on time step and on the interlayer. The physics is not qualitatively changed by these parameters. Stochastic uncertainties are approximately the size of the symbols.

Figure 2

(a) FN-DMC energies of different magnetic orderings considered for a $2\sqrt{2}\times 2\sqrt{2}\times 1$ unit cell of ${\mathrm{CaCuO}}_2$. (b) Hole charge density obtained by subtracting the charge density of the $x=0.125$ system from the $x=0.00$ antiferromagnetic ordering. (c) Spin density of the corresponding orderings for $x=0.125$. Both are calculated from the optimal single Slater determinant for that spin configuration for ${\mathrm{CaCuO}}_2$ and projected onto the $ab$ crystallographic plane. Correlations do not affect these pictures within statistical noise. In the density maps, blue is positive and red is negative. The density maps are normalized to the same value.

Figure 3

Optical gaps in FN-DMC and PBE0 calculated by promoting an electron in the Slater determinant, performed in the $2\times 2\times 1$ cell for $x=0.00$ and $x=0.25$, and in the $2\sqrt{2}\times 2\sqrt{2}\times 1$ cell for $x=0.125$. The twisted boundary condition of the wave function in the crystallographic primitive cell is labeled on the horizontal axis.

Figure 4

Orbitals contributing to the spin density of a two formula unit cell of ${\mathrm{CaCuO}}_2$. (a)–(c) Lower energy orbital that contributes to most of the spin density on the copper atoms. (d)–(f) Higher energy orbital that contributes to most of the spin density on the oxygen atoms. (g) Sum of the two spin densities, which accounts for most of the total spin density. Blue is positive, red is negative.

Figure 5

Charge (a) and spin (b) density in PBE0 for two holes in a $4\times 4\times 1$ supercell. The corresponding Fourier transforms for charge (c) and spin (d) have a Bragg peak at (0,0.125) because of the periodicity in addition to the plotted values. The $k$ values are in units of the reciprocal lattice of the primitive tetragonal crystallographic cell.