Lightly doped dimerized spin chain in the one-dimensional t-J-J' model

We study the one-dimensional t-J-J'-model in the limit of small hole doping $x$ and small $J/t$, $J'/t$. Special emphasis is put on the regime $J'/J \approx .5$ where a spin gap is present at small doping and the undoped spin chain is strongly dimerized. Using a perturbative approach and Luttinger liquid arguments, we demonstrate for this non-integrable class of models that the charge degrees of freedom behave as non-interacting spinless solitons in the dilute hole limit. Our approach is also used to evaluate the energy and mass renormalization of a single hole. Interestingly, the corrections of these quantities are in powers of $\sqrt{J/t}$. At $J'/J=.5$ we construct a variational spin-polaron wave function for the hole and find good agreement with our perturbative results.