Polaron states in a CuO chain

We introduce a one-dimensional model for a CuO chain, with holes and $S=1/2$ spins localized in $3d_{x^2-y^2}$ orbitals, and $p_\sigma$ oxygen orbitals without holes in the ground state. We consider a single hole doped at an oxygen site and study its propagation by spin-flip processes. We develop the Green's function method and treat the hole-spin coupling in the self-consistent Born approximation, similar to that successfully used to study polarons in the regular $t$-$J$ model. We present an analytical solution of the problem and investigate whether the numerical integration is a good approximation to this solution.