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

TL;DR

This paper reviews the non-linear sigma model approach to spin chains and ladders, generalizes Haldane's map, and presents new results on the phase diagram, spin gaps, and the existence of Haldane phases, supported by numerical fits.

Contribution

It introduces a generalized Haldane map for ladders, derives new theta parameter values, and proposes a finite size correction to the spin gap formula.

Findings

Even ladders are gapped, odd ladders are gapless.

Spin gap decreases exponentially with the number of legs.

Existence of a Haldane phase in two-legged ladders.

Abstract

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.