The transition between hole-pairs and four-hole clusters in four-leg tJ ladders

Holes weakly doped into a four-leg \tj ladder bind in pairs. At dopings exceeding a critical doping of $\delta_c\simeq {1/8}$ four hole clusters are observed to form in DMRG calculations. The symmetry of the ground state wavefunction does not change and we are able to reproduce this behavior qualitatively with an effective bosonic model in which the four-leg ladder is represented as two coupled two-leg ladders and hole-pairs are mapped on hard core bosons moving along and between these ladders. At lower dopings, $\delta<\delta_c$, a one dimensional bosonic representation for hole-pairs works and allows us to calculate accurately the Luttinger liquid parameter $\krho$, which takes the universal value $\krho=1$ as half-filling is approached.