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I prove that the Bethe roots describing either the ground state or a certainclass of "particle-hole" excited states of the XXZ spin-$1/2$ chain in anysector with magnetisation $\mathfrak{m} \in [0;1/2]$ exist and form, in theinfinite volume limit, a dense distribution on a subinterval of $\mathbb{R}$.The results holds for any value of the anisotropy $\Delta \geq -1 $. In fact, Iestablish an even stronger result, namely the existence of an all orderasymptotic expansion of the counting function associated with such roots. As acorollary, these results allow one to prove the existence and form of theinfinite volume limit of various observables attached to the model -theexcitation energy, momentum, the zero temperature correlation functions, so asto name a few- that were argued earlier in the literature.

On condensation properties of Bethe roots associated with the XXZ chain