Large N behavior of string solutions in the Heisenberg model

We investigate the large $N$ behavior of the complex solutions for the two magnon system in the S=1/2 Heisenberg XXZ model. The Bethe ansatz equations are numerically solved for the string solutions with new iteration method. A clear evidence of the violation of the string configurations is found at $N=22,62,121,200,299,417$, but the broken states are still Bethe states. The number of the Bethe states is consistent with the exact diagonalization, except one singular state.