Phenomenological Three-Orbital Spin-Fermion Model for Cuprates

A spin-fermion model that captures the charge-transfer properties of Cu-based high critical temperature superconductors is introduced and studied via Monte Carlo simulations. The strong Coulomb repulsion among $d$-electrons in the Cu orbitals is phenomenologically replaced by an exchange coupling between the spins of the itinerant electrons and localized spins at the Cu sites, formally similar to double-exchange models for manganites. This interaction induces a charge-transfer insulator gap in the undoped case (five electrons per unit cell). Adding a small antiferromagnetic Heisenberg coupling between localized spins reinforces the global tendency towards antiferromagnetic order. To perform numerical calculations the localized spins are considered classical, as in previous related efforts. In this first study, undoped and doped $8\times 8$ clusters are analyzed in a wide range of temperatures. The numerical results reproduce experimental features in the one-particle spectral function and the density-of-states such as $(i)$ the formation of a Zhang-Rice-like band with a dispersion of order $\sim 0.5$ eV and with rotational symmetry about wavevector $(\pi/2,\pi/2)$ at the top of the band, and $(ii)$ the opening of a pseudogap at the chemical potential upon doping. We also observed incipient tendencies towards spin incommensurability. This simple model offers a formalism intermediate between standard mean-field approximations, that fail at finite temperatures in regimes with short-range order, and sophisticated many-body techniques such as Quantum Monte Carlo, that suffer sign problems.