Analysis of the singlet-triplet model for the copper oxide plane within the paramagnetic state

The two-band Emery model has been reduced to an effective singlet-triplet model to describe the low-energy electronic properties of the ${\mathrm{CuO}}_2$ plane in oxide superconductors. The effective Hamiltonian is written in terms of Hubbard operators, projecting onto local singlet and triplet states of a doped hole. The projection method for two-time Green’s functions (GF’s) is applied to obtain the band structure and the density of states in the paramagnetic state. It is found that a singlet band that is mainly of oxygen character is located between the antibonding copper and the nonbonding oxygen bands. The triplet part in the singlet band has been found to be very small. This is due to the small mixing parameter and because the local singlet-triplet transition is forbidden due to time-reversal symmetry. The results are in good agreement with a random-phase-approximation-like decoupling for the GF’s of the original problem including the singlet operator [R. Hayn, Z. Phys. B 85, 169 (1991)]. With doping, a transfer of spectral weight occurs from the copper and the triplet to the singlet band.