Renormalization-Group Approach to a Three-Legs Fermionic Ladder

We study the spin and charge phase diagram of a three-legs ladder (at zero temperature) as a function of fermion density and of transverse single-particle hopping by means of a Renormalization-Group analysis rigorously controlled in the weak-coupling limit. Periodic boundary conditions in the direction transverse to the ladder produce frustrated magnetic excitations yielding a spin-gapped phase in a large region about and at half filling. Spin correlations are instead enhanced when open transverse boundary conditions are considered, yielding an ungapped phase in a wide region about half filling (as observed in the Sr$_{n-1}$Cu$_{n+1}$O$_{2n}$ stripes compounds) and at half filling down to a critical value of the transverse hopping $t_{\perp}$. At that critical value, the system undergoes a Mott transition at half filling by decreasing $t_{\perp}$.