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

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.

46 pages, V2 some clarifications added to Section 4

References in corpus (2)

• Thermal form-factor approach to dynamical correlation functions of integrable lattice models
• Deformed strings in the Heisenberg model

Cited by in corpus (9)

• Correlation functions of one-dimensional strongly interacting two-component gases
• A thermal form factor series for the longitudinal two-point function of the Heisenberg-Ising chain in the antiferromagnetic massive regime
• Statistical mechanics of integrable quantum spin systems
• Spin conductivity of the XXZ chain in the antiferromagnetic massive regime
• Thermal form-factor expansion of the dynamical two-point functions of local operators in integrable quantum chains
• Scaling in the massive antiferromagnetic XXZ spin-1/2 chain near the isotropic point
• Heisenberg XX chain, non-homogeneously parameterised generating exponential, and diagonally restricted plane partitions
• Fourth-neighbour two-point functions of the XXZ chain and the Fermionic basis approach
• Thermodynamics of an Ising-like XXZ chain in a longitudinal magnetic field in the framework of the Quantum Transfer Matrix approach