Spin-gap phase in the extended t-J chain

We study the one-dimensional deformed $t-t^{\prime}-J$ model in terms of the continuum field theories. We found that at low doping concentration and far away from the phase separation regime, there are two phases: the Luttinger liquid and the Luther-Emery liquid, depending on t'/t<(t'/t)_c or t'/t>(t'/t)_c, where (t'/t)_c>0. Moreover, the singlet superconducting correlations are dominant in the Luther-Emery liquid.