Phases from Lightly-Doped Two-Legged Spin Ladders

We study two-leg spin ladder systems analytically by using a t-J model which includes the interchain spin exchange and the interchain hopping term. The spin part is mapped to a quantum sine-Gordon model via a bosonization method that fully accounts for the phase fluctuation. The spin gap in the Luther-Emery phase evolves to gapless Luttinger liquid phases in even-ladders at certain doping concentration. This transition occurs as doping rate increases and/or the ratio of the interchain spin exchange and hopping to the intrachain couplings decreases. We also estimate the transition temperature at which the conventional electron phase is deconfined to spinon and holon.