Ground-state phase diagram of the extended two-leg t-J ladder

Inspired by the observation of a robust d-wave superconducting phase driven by tuning the next-nearestneighbor (NNN) electron hopping in recent density matrix renormalization group (DMRG) studies of sixand eight-leg t-J model, we systematically study the phase diagram of the two-leg t-J ladder with the NNN couplings (including NNN hopping and spin interaction) in a large region of doping level, by means of the DMRG calculations. Upon doping from half filling, we identify the Luther-Emery liquid (LEL) phase, which can be distinguished as the pairing-dominant and charge density-dominant regime by comparing the Luttinger parameter K rho. With the growing NNN couplings, pairing correlations are enhanced and correspondingly K rho increases, driving the system from the density-dominant to the pairing-dominant regime. In the TomonagaLuttinger liquid (TLL) phase in the larger doping region, we identify two TLL regimes with different features of charge density correlation. At the quarter filling (1/2 doping level), we find that the strong dimer orders of bond energy in the open system actually decay algebraically and thus do not indicate a spontaneous translational symmetry breaking. Our results show that in the LEL phase of the two-leg ladder, the NNN couplings seem to play the similar role as that on the wider t-J cylinder, and studies on this more accessible system can be helpful toward understanding the emergence of the remarkable d-wave superconducting phase on the wider system.