¡¡¡¡Over the past few years, considerable interest has focusedon systems of coupled chain coupling. Early theoretical studies of Heisenberg ladders (appropriate forthe strongly interacting, non-itinerant half-filled limit)revealed an interesting odd/even effect.1¨C5 If the numberof legs of the ladder, N, is even, the system is expectedto be a spin-liquid with a singlet ground stateand a (spin) gap to the lowest lying excitations carryingangular momentum. For odd N, the ground statehas quasi-long-range antiferromagnetic order and a set ofgapless spin-wave excitations, which puts it in the universalityclass of the single spin-1/2 Heisenberg chain.
¡¡¡¡Recent progress in the experimental preparation of relativelyisolated spin ladders has begun to probe some ofthis rich physics and appears to have verified these expectationsfor N = 2, 3.6¨C9The behavior of doped ladders, i.e. those with itinerantcharge carriers, is much richer. Particular theo reticalattention has been paid to the case N = 2, the twolegladder.2,3,10¨C16 Early motivation stemmed from thepossibility of realizing a concrete example of resonatingvalence bond (RVB) ideas.17,18 According to this line ofthought, since the two-leg Heisenberg ladder is a spin liquid,the doped carriers experience a short-range attractiveinteraction, leading to pairing and the persistence ofthe spin gap. Such behavior is indeed observed in simulationsof two-chain Hubbard19¨C21 and t-J models,2,22¨C25which push the current computational limits of numericalmethods working directly at zero temperature. Subsequentwork by numerous authors has since demonstratedthe existence of such a spin gap phase for low dopings bycontrolled analytical methods in weak Sleeve Coupling.13,14,11This weak-coupling approach has the additional advantagethat it provides a full picture of the phase diagram,even away from half-filling.
¡¡¡¡In this paper, we extend this analysis to more generalN-chain Hubbard models.26¨C29 Such an extension is usefulin two respects. First, it allows a determination of thephase diagram for any small value of N, thereby elucidatingthe physics of doped spin liquids, the even/odd effect,and geometrical frustration. Furthermore, our equationsallow a complete interpolation between one and two dimensions(along a particular path in parameter space¨C see below)¡£ An understanding of such a dimensionalcrossover21 is a crucial first step in interpreting experimentsin quasi-one-dimensional conductors.
¡¡¡¡To determine the behavior in the weak interactionlimit, we employ a generalization of the renormalizationgroup (RG) developed in Ref. 13 (the extension tothe particular case N = 3 has already by studied byArrigoni30,31)¡£ This provides a systematic basis for treatingthe logarithmic divergences arising in a naive perturbativeanalysis. Coupled with the technique of bosonization,the primary output of the RG is a ¡°gap¡± function describing pairing and the relative phase among thevarious spin and charge modes in the system. In thelimit of large N,  becomes identical to the gap functiondefined in the conventional BCS theory of superconductivity,and one may thereby connect our results directlywith higher dimensional analogues.
¡¡¡¡The RG also determines the zero temperature behavioras the chain coupling length is taken to infinity. Because sucha system is, for any finite N, still one-dimensional, itcannot sustain true off-diagonal long-range order, butis instead a generalized Luttinger liquid. The particularLuttinger liquid phase, within a general classificationscheme developed in Ref. 13 also follows from the gapfunction . We will use this notation, in which a phasewith m gapless charge and n gapless spin modes is denotedCmSn, in what follows.
¡¡¡¡The results of our calculations for positive U Hubbardchains in the phase diagrams are summarized in Figs.[6-10]. We emphasize that the phase diagrams are valid forarbitrary filling n and transverse hopping t¡Í except atsome specific lines (see below)¡£ First note the proliferationof phases as N increases from 2 to 4. We believethat this complexity persists even in the N ¡ú ¡Þ limit(but see below)¡£ The crossover to two dimensions is thushighly nontrivial. Secondly, despite the repulsive interactions,the majority of phases exhibit some degree ofreduction of gapless spin modes, i.e. pairing. The symmetryof the pair wavefunction is in most cases consistentwith a d-wave form. Unlike the two-chain case, however,as N is increased, gapless spin modes exist due to the presence of nodes in the pair wavefunction. Under different circumstances, Sleeve Coupling can be seen from Figs.[6-10], bothdx2?y2 and dxy states appear.