Haldane gap in the quasi one-dimensional nonlinear $\sigma$-model

This work studies the appearance of a Haldane gap in quasi one-dimensional antiferromagnets in the long wavelength limit, via the nonlinear $\sigma$-model. The mapping from the three-dimensional, integer spin Heisenberg model to the nonlinear $\sigma$-model is explained, taking into account two antiferromagnetic couplings: one along the chain axis ($J$) and one along the perpendicular planes ($J_\bot$) of a cubic lattice. An implicit equation for the Haldane gap is derived, as a function of temperature and coupling ratio $J_\bot/J$. Solutions to these equations show the existence of a critical coupling ratio beyond which a gap exists only above a transition temperature $T_N$. The cut-off dependence of these results is discussed.