Spin Gaps in Coupled t-J Ladders

Spin gaps in coupled $t$-$J$ ladders are investigated by exact diagonalization of small clusters up to 4$\times$8 sites. At half-filling, the numerical results for the triplet excitation spectrum are in very good agreement with a second order perturbation expansion in term of small inter-ladder and intra-ladder exchange couplings between rungs ($J/J^\prime$$<$$0.25$). The band of local triplet excitations moving coherently along the ladder (with momenta close to $\pi$) is split by the inter-ladder coupling. For intermediate couplings finite size scaling is used to estimate the spin gap. In the isotropic infinite 4-chain system (two coupled ladders) we find a spin gap of $0.245 J$, roughly half of the single ladder spin gap. When the system is hole doped, bonding and anti-bonding bound pairs of holes can propagate coherently along the chains and the spin gap remains finite.