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

Frank Göhmann, Salvish Goomanee, Karol K. Kozlowski, Junji Suzuki

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.

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- chain at finite temperature . Within the quantum inverse scattering method the physically pertinent observables at finite , 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- 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.

Paper Structure

This paper contains 23 sections, 17 theorems, 231 equations, 6 tables.

Table of Contents

  1. Introduction and main results
  2. Main results of the paper
  3. Notations
  4. Outline of the paper
  5. Traces of powers of the quantum transfer matrix at high temperatures
  6. Existence of a dominant Eigenvalue and commutativity of limits
  7. Bound on the dominant and subdominant Eigenvalues
  8. Commutativity of the limits
  9. The non-linear integral equation based description of the spectrum
  10. Preliminary discussion
  11. The formal infinite Trotter number limit
  12. Existence and uniqueness of solutions to the NLIE for large T
  13. An auxiliary problem grasping the leading high-temperature behaviour
  14. An auxiliary fixed point problem
  15. The free energy and correlation lengths at high temperatures
  16. ...and 8 more sections

Key Result

Theorem \oldthetheorem

There exist $T_0>0$ and $\epsilon>0$ such that for any $T>T_0$, the per-site free energy of the XXZ chain defined by definition energie libre admits the integral representation in which $\mathfrak{A}$ is the unique solution to the non-linear integral equation on $\mathcal{B}_{\mathfrak{c}}$ Here, the logarithm is defined as in which the integral runs from $\epsilon$ to $\xi \in \partial_{} \math

Theorems & Definitions (18)

  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Corollary \oldthetheorem
  • Proposition \oldthetheorem
  • Proposition \oldthetheorem
  • Lemma 3.1
  • Theorem \oldthetheorem
  • ...and 8 more