Spin and charge modes of thet-Jladder

The spin and charge excitations of the t-J ladder are studied by exact diagonalization techniques for several electron densities. The various modes are classified according to their spin (singlet or triplet excitations) and their parity under a reflection with respect to the symmetry axis along the chains and a finite-size scaling of the related gaps is performed. At low doping, formation of hole pairs leads to a spin gap for all J/t ratios. This phase is characterized by (at least) one vanishing energy mode only in the charge bonding channel when Kₓ{→}0 consistent with the existence of superconducting pairing correlations. At larger doping the spin gap disappears. Although the antibonding modes remain gapped, low-energy Kₓ{~}0 and Kₓ{~}2kF spin and charge bonding modes are found consistent with a single-band Luttinger scenario. At sufficient low electron density and above a critical value of J/t we also expect another phase of electron pairs with gapped spin excitations.