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

Abstract

We study the dimensional crossover from 1D to 2D in half-filled and lightly doped, weakly interacting N-leg Hubbard ladders. In this case, the Hubbard ladders are equivalent to a 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}\propto exp[-a exp(c N)] (a<<1 and c\sim 1), i.e., below E_{c}, the interactions take place only 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 2D-like (insulating) antiferromagnet; we obtain an analytical expression for the Hamiltonian which is similar to the 2D Heisenberg antiferromagnet. Bosonization techniques show, that the Mott insulator is as well below and above E_{c} of the same type as in N/2 half-filled two-leg ladders; the charge-sector is therefore the same in 1D and 2D. Doping away from half-filling, we find that the effect of an increasing doping is very similar to decreasing the number of legs N: In both cases interactions between unpaired bands are suppressed and the antiferromagnetic correlations are reduced. The resulting band pairs (j,N+1-j) are then insulating spin-liquids and when doped, there is a spin-gap, but phase coherence exists only within pairs of Fermi momenta (k_{Fj},pi-k_{Fj}) and (pi-k_{Fj},k_{Fj}). At higher doping levels delta_{c}=delta_{c}(N), phase coherence between all bands sets in and the system becomes a 2D-like d-wave superconductor (delta_{c} goes to 0 in the 2D limit).