Weakly coupled Hubbard chains at half-filling and confinement

We study two (very) weakly coupled Hubbard chains in the half-filled case, and especially the situation where the intrachain Mott scale m is much larger than the (bare) single-electron interchain hopping t_⊥. First, we find that the divergence of the intrachain umklapp channel at the Mott transition results in the complete vanishing of the single-electron interchain hopping: this is significant of a strong confinement of coherence along the chains. Excitations are usual charge fermionic solitons and spinon-(anti)spinon pairs of the Heisenberg chain. Then, we show rigorously how the tunneling of spinon-(anti)spinon pairs produces an antiferromagnetic interchain exchange of the order of J_⊥ = t_⊥²/m. In the "confined" phase and in the far infrared, the system behaves as a pure spin ladder. The final result is an insulating ground state with spin-gapped excitations exactly as in the opposite "delocalized" limit (i.e., for rather large interchain hoppings) where the two-leg ladder is in the well-known insulating D-Mott phase. Unlike materials with an infinite number of coupled chains (Bechgaard salts), the confinement/deconfinement transition at absolute zero is here a simple crossover: no metallic phase is found in undoped two-leg ladders. This statement might be generalized for N-leg ladders with N = 3,4,... (but not too large).