Rung-rung current correlations on a 2-legt−Jladder

We report the results of numerical calculations of rung-rung current correlations on a 2-leg $t-J$ ladder with $J/t=0.35\mathrm{}$ for dopings $x=0.125\mathrm{}$ and $x=\mathrm{0.19.}\mathrm{}$ We find that the amplitude of these correlations decays exponentially. We argue that this can be understood within a bosonization framework in terms of the pinned phase variables associated with a C1SO phase with $d_{x^2-y^2}$-like power law pairing correlations. We point out that coexistence of power law pairing correlations and orbital current correlations in any 2-leg ladder model requires the existence of at least two gapless boson fields.