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

physical review. b, condensed matter

Transition between hole pairs and four-hole clusters in four-leg<i>t</i>-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>J</mml:mi></mml:math>ladders

Transition between hole pairs and four-hole clusters in four-legt-Jladders