On condensation properties of Bethe roots associated with the XXZ chain

TL;DR

This paper proves the existence and density properties of Bethe roots for the XXZ spin chain across various states and parameters, enabling the rigorous analysis of physical observables in the infinite volume limit.

Contribution

It establishes the existence, density, and asymptotic expansion of Bethe roots for the XXZ chain for all anisotropy values, extending previous results and enabling analysis of physical quantities.

Findings

Bethe roots form dense distributions in the infinite volume limit.

Existence of all-order asymptotic expansion of the counting function.

Rigorous derivation of infinite volume limits for physical observables.

Abstract

I prove that the Bethe roots describing either the ground state or a certain class of "particle-hole" excited states of the XXZ spin-$1/2$ chain in any sector with magnetisation $\mathfrak{m} \in [0;1/2]$ exist and form, in the infinite 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, I establish an even stronger result, namely the existence of an all order asymptotic expansion of the counting function associated with such roots. As a corollary, these results allow one to prove the existence and form of the infinite volume limit of various observables attached to the model -the excitation energy, momentum, the zero temperature correlation functions, so as to name a few- that were argued earlier in the literature.