Non-equivalence between Heisenberg XXZ spin chain and Thirring model

. The Bethe ansatz equations for the spin 1/2 Heisenberg XXZ spin chain are numerically solved, and the energy eigenvalues are determined for the antiferromagnetic case. We examine the relation between the XXZ spin chain and the massless Thirring model, and show that the spectrum of the XXZ spin chain has a gapless excitation while the regularized Thirring model calculated with the Bogoliubov transformation method has a finite gap. This finite gap spectrum is also confirmed by the Bethe ansatz solution of the massless Thirring model.