Thermodynamics of the spin-$1/2$ Heisenberg-Ising chain at high temperatures: a rigorous approach

TL;DR

This paper rigorously proves key conjectures about the high-temperature behavior of the Heisenberg-Ising spin-1/2 chain, including integral representations of observables and asymptotic expansions of eigenvalues, advancing theoretical understanding of quantum spin chains.

Contribution

It provides a rigorous proof of conjectures related to integral representations and eigenvalues for the Heisenberg-Ising chain at high temperatures, previously only heuristically assumed.

Findings

Proved existence and uniqueness of solutions to auxiliary integral equations at high T.

Derived large-T asymptotic expansion for sub-dominant eigenvalues.

Validated integral representations for finite-temperature observables.

Abstract

This work develops a rigorous setting allowing one to prove several features related to the behaviour of the Heisenberg-Ising (or XXZ) spin-$1/2$ chain at finite temperature $T$. Within the quantum inverse scattering method the physically pertinent observables at finite $T$, such as the \textit{per}-site free energy or the correlation length, have been argued to admit integral representations whose integrands are expressed in terms of solutions to auxiliary non-linear integral equations. The derivation of such representations was based on numerous conjectures: the possibility to exchange the infinite volume and the infinite Trotter number limits, the existence of a real, non-degenerate, maximal in modulus Eigenvalue of the quantum transfer matrix, the existence and uniqueness of solutions to the auxiliary non-linear integral equations, as well as the possibility to take the infinite Trotter number limit on their level. We rigorously prove all these conjectures for temperatures large enough. As a by product of our analysis, we obtain the large-$T$ asymptotic expansion for a subset of sub-dominant Eigenvalues of the quantum transfer matrix and thus of the associated correlation lengths. This result was never obtained previously, not even on heuristic grounds.