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 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.

中文翻译：

---

高温下自旋 1/2 海森堡-伊辛链的热力学：严格的方法

这项工作开发了一个严格的设置，允许人们证明与 Heisenberg-Ising（或 XXZ）自旋 1/2 链在有限温度 T 下的行为相关的几个特征。在量子逆散射方法中，有限 T 处的物理相关可观测量，例如每个位点的自由能或相关长度，已被认为允许积分表示，其被积函数以辅助非线性积分方程的解表示。此类表示的推导基于许多猜想：交换无限体积和无限 Trotter 数限制的可能性、存在实数、非退化、最大模量的量子传递矩阵的特征值、存在和唯一性辅助非线性积分方程的解，以及在他们的级别上采取无限的猪蹄数量限制的可能性。对于足够大的温度，我们严格证明了所有这些猜想。作为我们分析的副产品，我们获得了量子传递矩阵的次显性特征值的子集的大 T 渐近展开式，从而获得了相关联的相关长度。这个结果以前从未获得过，即使是基于启发式的。