Dimensional crossover in half-filled and lightly doped N-leg Hubbard ladders

We study the dimensional crossover from one to two dimensions in half-filled and lightly doped, weakly interacting N-leg Hubbard ladders. In this case, the Hubbard ladders are equivalent to an N-band model. Using renormalization-group techniques, we find that the half-filled ladders exhibit (in the spin sector) an odd-even effect only below a crossover energy E_c∝exp[-αexp(γN)] (α≪1 and γ∼1 depend on the interaction strength and on the hopping matrix elements): Below E_c, the dominant interactions take place within band pairs (j,N+1-j) [and within the band (N+1)/2 for N odd], such that even-leg ladders are an insulating spin liquid, while odd-leg ladders have one gapless spinon mode. In contrast, above the energy scale E_c, all bands are interacting with each other and the system is a two-dimensional- (2D)-like (insulating) antiferromagnet; we obtain an analytical expression for the Hamiltonian which is similar to the 2D Heisenberg antiferromagnet. Bosonization techniques show that in the charge sector the Mott insulator is as well below and above E_c of the same type as in N/2 half-filled two-leg ladders. Doping away from half-filling, we find that the effect of an increasing doping δ is very similar to decreasing the number of legs N: In both cases interactions between unpaired bands are suppressed and thereby the antiferromagnetic correlations reduced. The resulting band pairs form then insulating spin liquids and when doped, there is a spin gap, but phase coherence exists only within the band pairs. At higher doping levels δ_c=δ_c(N), phase coherence between all band pairs sets in and the system becomes a 2D-like d-wave superconductor [δ_c(N)→0 for N⃗∞].