I am wondering whether it is known, or whether it can easily seen from the Bethe ansatz solution, what the gap of the spin-1/2 XXZ model of finite size $N$ with periodic boundary conditions ($H=\sum\sigma^x_i\sigma^x_{i+1}+\sigma^y_i\sigma^y_{i+1}+\Delta\sigma^z_i\sigma^z_{i+1}$) is in a specific magnetisation sector (that is $\frac{1}{N}\sum\sigma^z_i$ is fixed to some value $m$)? Alternatively, what is the scaling of the gap in the limit $N \rightarrow \infty$? I am in interested in the paramagnetic and the ferromagnetic phases of the model, that is $-1 < \Delta$.
