Spin gap in coupled quarter-filled Hubbard ladders

We present DMRG calculations in quarter-filled ladder systems coupled to the lattice. Single and coupled ladders are considered, with model parameters obtained from first-principles band-structure calculations for $\alpha^\prime$-$NaV_2O_5$. The relevant Holstein coupling to the lattice causes zig-zag lattice distortions, concurrently with the formation of a charge ordered-state. The coupling to the lattice drastically reduces the critical nearest neighbor Coulomb repulsion $V$ and changes the critical exponent of the corresponding quantum phase transition. We find excellent agreement with experimental data for lattice distortion, charge gap, and charge order parameter at a $V^*$ which agrees with previous estimates. The spin gap extrapolates to zero on a single ladder and on two coupled ladders, in spite of a dimerization of spins in chain direction. A finite spin gap appears only on a system with periodicity of {\em four} ladders, as observed experimentally in $\alpha^\prime$-$NaV_2O_5$. The gap is of the right size at $V^*$ but, surprisingly, vanishes again at larger charge order.