On the application of the non-linear sigma model to spin chains and spin ladders

We review the non-linear sigma model approach (NLSM) to spin chains and spin ladders, presenting new results. The generalization of the Haldane's map to ladders in the Hamiltonian approach, give rise to different values of the theta parameter depending on the spin S, the number of legs ne and the choice of blocks needed to built up the NLSM fields. For rectangular blocks we obtain theta = theta or 2 pi S depending on wether n(l), is even or odd, while for diagonal blocks we obtain theta = 2 pi Sn-l. Both results agree module 2 pi, and yield the same prediction, namely that even ( resp. odd) ladders are gapped (resp. gapless). For even leeged ladders we show that the spin gap collapses exponentially with ne and we propose a finite size correction to the gap formula recently derived by Chakravarty using the 2+1 NSLM, which gives a good fit of numerical results. We show the existence of a Haldane phase in the two legged ladder using diagonal blocks and finally we consider the phase diagram of dimerized ladders.