Parent insulating state

The parent compounds of cuprate superconductors contain CuO2 planes, in which Cu$^{2+}$ ions have one hole in $d_{x^2 - y^2}$ orbitals. This implies a half-filled band, and for non-interacting electrons would correspond to a metallic state. However, due to electron-electron interactions, below the Neel temperature, the parent compounds become antiferromagnetic, with a gap in the electronic excitation spectrum and a characteristic magnetic excitations spectrum.

A system that is insulating due to Coulomb repulsion among electrons within a single
band is commonly labeled a Mott insulator. This categorization is often applied to
the cuprate parent compounds; however, they are more accurately described as charge-transfer insulators. The distinction between Mott insulators and charge transfer insulators is based upon comparison of Mott gap $U$ and charge transfer gap $\Delta$ ^[1]. In the context of cuprates, $U$ corresponds to the energy cost associated with moving an electron from one Cu ion to another (cost of double occupancy), while $\Delta$ is the cost of moving an electron from O ion to Cu. Since for cuprates $\Delta < U$, they belong to the charge transfer insulator class.

A cubic crystal field causes a splitting of the Cu 3d
orbitals into two groups: $e_g$ symmetry ($x^2-y^2$, $3z^2 - r^2$ and $t_{2g}$ ($xy$, $xz$, $yz$), with $e_g$ being at higher energy. With a tetragonal elongation of the CuO6 octahedra, the $3z^2-r^2$ orbital is lowered in energy relative to the $x^2-y^2$, so that the one hole sits in the latter orbital. It has recently become possible to determine the energy splittings between Cu 3d orbitals with resonant inelastic x-ray scattering at the Cu L3 edge ^[2]. Recent ab
initio calculations are in good agreement with the measurements ^[3]. For La2CuO4,
measurements show that the d-d excitation energies from the $x^2-y^2$ state are 1.7 eV to $3z^2-r^2$, 1.8 eV to $xy$, and 2.1 eV to $xz$, $yz$.

Early ab initio calculations yielded good results for the electronic spectra at high energies (> 5 eV) but were predicting metallic conductivity at low frequencies ^[4], ^[5].

References

1. J. Zaanen, G. A. Sawatzky, and J. W. Allen, Phys. Rev. Lett. 55, 418–421 (1985).
2. M. M. Sala, V. Bisogni, C. Aruta, G. Balestrino, H. Berger, N. B. Brookes, G. M. de Luca, D. D. Castro, M. Grioni, M. Guarise, P. G. Medaglia, F. M. Granozio, M. Minola, P. Perna, M. Radovic, M. Salluzzo, T. Schmitt, K. J. Zhou, L. Braicovich, and G. Ghiringhelli, New J. Phys. 13, 043026 (2011).
3. L. Hozoi, L. Siurakshina, P. Fulde, and J. van den Brink, Sci. Rep. 1, doi:10.1038/srep00065 (2011).
4. I. I. Mazin, E. G. Maksimov, S. N. Rashkeev, S. Y. Savrasov, and Y. A. Uspenskii, JETP Lett. 47, 113 (1988).
5. S. Uchida, T. Ido, H. Takagi, T. Arima, and Y. Tokura, Phys. Rev. B 43, 7942 (1991).