On the Application of the Non Linear Sigma Model to Spin Chains and Spin Ladders

Details On the Application of the Non Linear Sigma Model to Spin Chains and Spin Ladders On the Application of the Non Linear Sigma Model to Spin Chains and Spin Ladders

1996-10-08

arxiv.org/abs/cond-mat/9610057v2

Lectures Notes in Physics, vol.478, eds. G. Sierra , and M.A. Martin-Delgado, Springer-Verlag (1997).

Physics

Condensed Matter

Statistical Mechanics

25 pages, Latex, 7 figures in postscript files, Proc. of the 1996 El Escorial Summer School on "Strongly Correlated Magnetic a

Scientific paper

10.1007/BFb0104637

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 $n_{\ell}$ and the choice of blocks needed to built up the NLSM fields. For rectangular blocks we obtain $\theta = 0 $ or $2 \pi S$ depending on wether $n_{\ell}$, is even or odd, while for diagonal blocks we obtain $\theta = 2 \pi S n_{\ell}$. Both results agree modulo $2 \pi$, and yield the same prediction, namely that even ( resp. odd) ladders are gapped (resp. gapless). For even legged ladders we show that the spin gap collapses exponentially with $n_{\ell}$ 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.

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