Exact regimes of collapsed and extra two-string solutions in the two down-spin sector of the spin-1/2 massive XXZ spin chain

We derive exactly the number of complex solutions with two down-spins in the massive regime of the periodic spin-1/2 XXZ spin chain of $N$ sites. Here we remark that every solution of the Bethe ansatz equations is characterized by a set of quantum numbers. We derive them analytically for all the complex solutions in the sector, which we call two-string solutions. We show that in a region of $N$ and $\Delta$ the number of two-string solutions is by two larger than the number due to the string hypothesis, i.e., an extra pair of two-strings appears. We determine it exactly and also such regions where $m$ two-string solutions collapse for any positive integers $m$. We illustrate the extra and standard two-string solutions numerically. In the sector we show that the string deviations are exponentially small with respect to $N$ if $N$ is large. We argue that for any finite solution of the spin-1/2 XXX chain there is such a solution of the spin-1/2 XXZ chain that has the same quantum numbers in common with the XXX solution.