Confinement and transverse conductivity in coupled Luttinger liquids

One--particle interchain hopping in a system of coupled Luttinger liquids is investigated by use of exact diagonalizations techniques. Firstly, the two chains problem of spinless fermions is studied in order to see the behaviour of the band splitting as a function of the exponent $\alpha$ which characterizes the $1D$ Luttinger liquid. Moderate intra-chain interactions can lead to a strong reduction of this splitting. The on-set of the confinement within the individual chains (defined by a vanishing splitting) seems to be governed by $\alpha$. We give numerical evidence that inter-chain coherent hopping can be totally suppressed for $\alpha\sim 0.4$ or even smaller $\alpha$ values. The transverse conductivty is shown to exhibit a strong incoherent part. Even when coherent inter-chain hopping is believed to occur (at small $\alpha$ values), it is shown that the coherent Drude weight is always significantly smaller than the incoherent weight. Implications for the optical experiments in quasi-1D organic or high-$T_c$ superconductors is outlined.