Thermodynamics of the spin-$1/2$ Heisenberg-Ising chain at high temperatures: a rigorous approach
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
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2018
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| _version_ | 1866916287577325568 |
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| author | Göhmann, Frank Goomanee, Salvish Kozlowski, Karol K. Suzuki, Junji |
| author_facet | Göhmann, Frank Goomanee, Salvish Kozlowski, Karol K. Suzuki, Junji |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1811_12020 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Thermodynamics of the spin-$1/2$ Heisenberg-Ising chain at high temperatures: a rigorous approach Göhmann, Frank Goomanee, Salvish Kozlowski, Karol K. Suzuki, Junji Mathematical Physics Strongly Correlated Electrons Exactly Solvable and Integrable Systems 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. |
| title | Thermodynamics of the spin-$1/2$ Heisenberg-Ising chain at high temperatures: a rigorous approach |
| topic | Mathematical Physics Strongly Correlated Electrons Exactly Solvable and Integrable Systems |
| url | https://arxiv.org/abs/1811.12020 |