  We study the impact of next-nearest-neighbor (nnn) hopping on the low-energy
collective excitations of strongly correlated doped antiferromagnetic cuprate
spin chains. Specifically, we use exact diagonalization and the density matrix
renormalization group method to study the single-particle spectral function,
the dynamical spin and charge structure factors, and the Cu $L$-edge resonant
inelastic x-ray scattering (RIXS) intensity of the doped $t$-$t^\prime$-$J$
model for a set of $t^\prime$ values. We find evidence for the breakdown of
spin-charge separation as $|t^\prime|$ increases and identify its fingerprints
in the dynamical response functions. The inclusion of nnn hopping couples the
spinon and holon excitations, resulting in the formation of a spin-polaron,
where a ferromagnetic spin polarization cloud dresses the doped carrier. The
spin-polaron manifests itself as additional spectral weight in the dynamical
correlation functions, which appear simultaneously in the spin- and
charge-sensitive channels. We also demonstrate that RIXS can provide a unique
view of the spin-polaron, due to its sensitivity to both the spin and charge
degrees of freedom.
