Opening of the Haldane gap in anisotropic two- and four-leg spin ladders

Abstract

We study the opening of the Haldane gap in two-leg and four-leg anisotropic spin ladders using bosonization and renormalization group methods, and we determine the phase diagram as a function of the interchain coupling and relative anisotropy ${J}^{z}{/J}^{\mathrm{xy}}.$ It is found that the opening of the Haldane gap is qualitatively different for the two cases considered. For the two-leg ladder the Haldane gap opens for arbitrarily small interchain coupling, independent of ${J}^{z}{/J}^{\mathrm{xy}},$ and the Haldane phase exists in a large region of parameter space. For the four-leg ladder the opening of the Haldane gap is strongly dependent on both the interchain coupling as well as ${J}^{z}{/J}^{\mathrm{xy}},$ and the Haldane phase exists only in a narrow region about the isotropic antiferromagnet.