We report on the closing of gaps in the ground state of the critical Heisenberg-Ising chain at momentum $π$. For half-filling, the gap closes at special values of the anisotropy $Δ= \cos(π/Q)$, $Q$ integer. We explain this behavior with the help of the Bethe Ansatz and show that the gap scales as a power of the system size with variable exponent depending on $Δ$. We use a finite-size analysis to calculate this exponent in the critical region, supplemented by perturbation theory at $Δ\sim 0$. For rational $1/r$ fillings, the gap is shown to be closed for {\em all} values of $Δ$ and the corresponding perturbation expansion in $Δ$ shows a remarkable cancellation of various diagrams.

12 RevTeX pages + 4 figures upon request