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Thermodynamics of the Spin-1/2 Heisenberg–Ising Chain at High Temperatures: a Rigorous Approach

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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 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.

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Notes

  1. While we make the choice of such a prescription, it is not so relevant in that the choice of any boundary value would still lead to the same definition of \(\widehat{\mathfrak {a}}\).

  2. In principle, degeneracies of \(|\widehat{\Lambda }_a|\) may occur and then the choice of the ordering for the various Eigenvalues with fixed modulus is taken in the direction of increasing arguments, with \(\mathrm {arg} \in [ -\pi \,; \pi [ \).

  3. The class \(\mathcal {C}^{\epsilon }_{\alpha , \varrho }\) is defined in Definition 5.1.

  4. To be precise, they are slightly away from \(\pm \mathrm i\); they are at \(\pm 1.00000000629\cdots \mathrm i\).

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Acknowledgements

The authors would like to thank Patrick Dorey, Andreas Klümper and Eric Vernier for stimulating discussions. The work of FG was supported by the Deutsche Forschungsgemeinschaft within the framework of the research unit FOR 2316. The work of SG and KKK was supported by the CNRS, Projet international de coopération scientifique No. PICS07877: Fonctions de corrélations dynamiques dans la chaîne XXZ à température finie, Allemagne, 2018–2020. JS was supported by JSPS KAKENHI Grants Numbers 15K05208, 18K03452 and 18H01141.

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Authors and Affiliations

  1. Fakultät für Mathematik und Naturwissenschaften, Bergische Universität Wuppertal, 42097, Wuppertal, Germany

    Frank Göhmann

  2. CNRS, Laboratoire de Physique, Univ Lyon, ENS de Lyon, Univ Claude Bernard Lyon 1, 69342, Lyon, France

    Salvish Goomanee & Karol K. Kozlowski

  3. Department of Physics, Shizuoka University, Ohya, 836, Shizuoka, Japan

    Junji Suzuki

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  1. Frank Göhmann
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  2. Salvish Goomanee
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Correspondence to Karol K. Kozlowski.

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Göhmann, F., Goomanee, S., Kozlowski, K.K. et al. Thermodynamics of the Spin-1/2 Heisenberg–Ising Chain at High Temperatures: a Rigorous Approach. Commun. Math. Phys. 377, 623–673 (2020). https://doi.org/10.1007/s00220-020-03749-6

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