Gap's in the antiferromagnetic Heisenberg model

We study the one-dimensional spin-1/2 antiferromagnetic Heisenberg model exposed to an external field, which is a superposition of a homogeneous field $h_{3}$ and a small periodic field of strength $h_{1}$. For the case of a transverse staggered field a gap opens, which scales with $h_{1}^{\epsilon_{1}}$, where $\epsilon_{1}=\epsilon_{1}(h_{3})$ is given by the critical exponent $\eta_{1}(M(h_{3}))$ defined through the transverse structure factor of the model at $h_{1}=0$. For the case of a longitudinal periodic field with wave vector $q=\pi/2$ and strength $h_{q}$ a plateau is found in the magnetization curve at $M=1/4$. The difference of the upper- and lower magnetic field scales with $h_{3}^{u}-h_{3}^{l}\sim h_{q}^{\epsilon_{3}}$, where $\epsilon_{3}=\epsilon_{3}(h_{3})$ is given by the critical exponent $\eta_{3}(M(h_{3}))$ defined through the longitudinal structure factor of the model at $h_{q}=0$.