We study the effect of doping away from half-filling in weakly (but finitely) interacting N-leg Hubbard ladders using renormalization-group and bosonization techniques. For a small on-site repulsion U, the N-leg Hubbard ladders are equivalent to an N-band model, where at half-filling the Fermi velocities are ${v}_{1}{=v}_{N}<{v}_{2}{=v}_{N\ensuremath{-}1}<\ensuremath{\cdot}\ensuremath{\cdot}\ensuremath{\cdot}.$ We then obtain a hierarchy of energy scales, where the band pairs $(j,N+1\ensuremath{-}j)$ are successively frozen out. The low-energy Hamiltonian is then the sum of
N/2 [or $(N\ensuremath{-}1)/2$ for N odd] two-leg ladder Hamiltonians without gapless excitations (plus a single chain for N odd with one gapless spin mode)---similar to the N-leg Heisenberg spin ladders. The energy scales lead to a hierarchy of gaps. Upon doping away from half-filling, the holes first enter the band(s) with the smallest gap: For odd N, the holes enter first the nonbonding band
(N+1)/2 and the phase is a Luttinger liquid, while for even N, the holes enter first the band pair
(N/2,N/2+1) and the phase is a Luther-Emery liquid, similar to numerical treatments of the $t\ensuremath{-}J$ model, i.e., at and close to half-filling, the phases of the Hubbard ladders for small and large U are the same. For increasing doping, hole pairs subsequently enter at critical dopings the other band pairs $(j,N+1\ensuremath{-}j)$ (accompanied by a diverging compressibility): The Fermi surface is successively opened by doping, starting near the wave vector $(\ensuremath{\pi}/2,\ensuremath{\pi}/2).$ Explicit calculations are given for the cases
N=3 and 4.
We consider electrons on a two-leg triangular ladder at half filling and in an orbital magnetic field. In a two-band regime in the absence of the field, the electronic system remains conducting for weak interactions since there is no four-fermion umklapp term. We find that in the presence of the orbital field there is a four-fermion umklapp and it is always relevant for repulsive interactions. Thus in this special ladder, the combination of the orbital magnetic field and interactions provides a mechanism to drive metal-insulator transition already at weak coupling. We discuss properties of the possible resulting phases C0S2 and various C0S1 and C0S0.
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 ${E}_{c}\ensuremath{\propto}\mathrm{exp}[\ensuremath{-}\ensuremath{\alpha}\mathrm{exp}(\ensuremath{\gamma}N)]$ $(\ensuremath{\alpha}\ensuremath{\ll}1$ and $\ensuremath{\gamma}\ensuremath{\sim}1$ depend on the interaction strength and on the hopping matrix elements): Below
Ec, the dominant interactions take place within band pairs $(j,N+1\ensuremath{-}j)$ [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
Ec, 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
Ec 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 $\ensuremath{\delta}$ 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 ${\ensuremath{\delta}}_{c}={\ensuremath{\delta}}_{c}(N),$ phase coherence between all band pairs sets in and the system becomes a 2D-like d-wave superconductor $[{\ensuremath{\delta}}_{c}(N)\ensuremath{\rightarrow}0$ for $\stackrel{\ensuremath{\rightarrow}}{N}\ensuremath{\infty}].$
We study a two-leg fermionic Hubbard ladder model with a state-dependent hopping. We find that, contrary to the case without a state-dependent hopping, for which the system has a superfluid nature regardless of the sign of the interaction at incommensurate filling, in the presence of such a hopping a spin-triplet superfluid, spin-density wave, and charge-density wave phases emerge. We examine our results in the light of recent experiments on periodically driven optical lattices in cold atoms. We give protocols allowing us to realize the spin-triplet superfluid elusive in the cold atoms.
We study the pair correlations of a spin-imbalanced two-leg ladder with attractive interactions, using the density matrix renormalization group method. We identify regions in the phase diagram spanned by the chemical potential and the magnetic field that can harbor Fulde-Ferrell-Larkin-Ovchinnikov- (FFLO-)like physics. Results for the pair structure factor, exhibiting multiple pairing wave vectors, substantiate the presence of FFLO-like correlations. We further discuss phase separation scenarios induced by a harmonic trap, which differ from the case of isolated chains.