Effective one-band models for the 1D cuprate Ba$_{2-x}$Sr$_x$CuO$_{3+\delta}$

We consider a multiband Hubbard model $H_m$ for Cu and O orbitals in Ba$_{2-x}$Sr$_x$CuO$_{3+\delta}$ similar to the tree-band model for two-dimensional (2D) cuprates. The hopping parameters are obtained from maximally localized Wannier functions derived from \textit{ab initio} calculations. Using the cell perturbation method, we derive both a generalized $t-J$ model $H_{tJ}$ and a one-band Hubbard model $H_{H}$ to describe the low-energy physics of the system. $H_{tJ}$ has the advantage of having a smaller relevant Hilbert space, facilitating numerical calculations, while additional terms should be included in $H_{H}$ to accurately describe the multi-band physics of $H_m$. Using $H_{tJ}$ and DMRG, we calculate the wave-vector resolved photoemission and discuss the relevant features in comparison with recent experiments. In agreement with previous calculations, we find that the addition of an attractive nearest-neighbor interaction of the order of the nearest-neighbor hopping shifts weight from the $3 k_F$ to the holon-folding branch. Kinetic effects also contribute to this process.