Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 10 (2014), 043, 18 pages      arXiv:1311.6959
Contribution to the Special Issue in honor of Anatol Kirillov and Tetsuji Miwa

Functions Characterizing the Ground State of the XXZ Spin-1/2 Chain in the Thermodynamic Limit

Maxime Dugave a, Frank Göhmann a and Karol Kajetan Kozlowski b
a) Fachbereich C - Physik, Bergische Universität Wuppertal, 42097 Wuppertal, Germany
b) IMB, UMR 5584 du CNRS, Université de Bourgogne, France

Received November 28, 2013, in final form April 07, 2014; Published online April 11, 2014

We establish several properties of the solutions to the linear integral equations describing the infinite volume properties of the XXZ spin-1/2 chain in the disordered regime. In particular, we obtain lower and upper bounds for the dressed energy, dressed charge and density of Bethe roots. Furthermore, we establish that given a fixed external magnetic field (or a fixed magnetization) there exists a unique value of the boundary of the Fermi zone.

Key words: linear integral equations; quantum integrable models; dressed quantities.

pdf (438 kb)   tex (28 kb)


  1. Bazhanov V.V., Lukyanov S.L., Zamolodchikov A.B., Integrable structure of conformal field theory. II. Q-operator and DDV equation, Comm. Math. Phys. 190 (1997), 247-278, hep-th/9604044.
  2. Bethe H., Zur Theorie der Metalle: Eigenwerte und Eigenfunktionen der linearen Atomkette, Z. Phys. 71 (1931), 205-226.
  3. Boos H., Jimbo M., Miwa T., Smirnov F., Hidden Grassmann structure in the XXZ model IV: CFT limit, Comm. Math. Phys. 299 (2010), 825-866, arXiv:0911.3731.
  4. de Vega H.J., Woynarovich F., Method for calculating finite size corrections in Bethe ansatz systems: Heisenberg chain and six-vertex model, Nuclear Phys. B 251 (1985), 439-456.
  5. des Cloizeaux J., Gaudin M., Anisotropic linear magnetic chain, J. Math. Phys. 7 (1966), 1384-1400.
  6. des Cloizeaux J., Pearson J.J., Spin-wave spectrum of the antiferromagnetic linear chain, Phys. Rev. 128 (1962), 2131-2135.
  7. Destri C., de Vega H.J., Unified approach to thermodynamic Bethe ansatz and finite size corrections for lattice models and field theories, Nuclear Phys. B 438 (1995), 413-454, hep-th/9407117.
  8. Dorlas T.C., Samsonov M., On the thermodynamic limit of the 6-vertex model, arXiv:0903.2657.
  9. Dugave M., Göhmann F., Kozlowski K.K., Thermal form factors of the XXZ chain and the large-distance asymptotics of its temperature dependent correlation functions, J. Stat. Mech. Theory Exp. 2013 (2013), P06002, 52 pages, arXiv:1305.0118.
  10. Dugave M., Göhmann F., Kozlowski K.K., Low-temperature large-distance asymptotics of the transversal two-point functions of the XXZ chain, arXiv:1401.4132.
  11. Essler F.H.L., Frahm H., Göhmann F., Klümper A., Korepin V.E., The one-dimensional Hubbard model, Cambridge University Press, Cambridge, 2005.
  12. Gaudin M., Thermodynamics of a Heisenberg-Ising ring for $\Delta \geq 1$, Phys. Rev. Lett. 26 (1971), 1301-1304.
  13. Griffiths R.B., Magnetization curve at zero temperature for the antiferromagnetic Heisenberg linear chain, Phys. Rev. 133 (1964), 768-775.
  14. Hulthén L., Über das Austauschproblem eines Kristalles, Arkiv Mat. Astron. Fys. A 26 (1938), 1-106.
  15. Kitanine N., Kozlowski K.K., Maillet J.M., Slavnov N.A., Terras V., Algebraic Bethe ansatz approach to the asymptotic behavior of correlation functions, J. Stat. Mech. Theory Exp. 2009 (2009), P04003, 66 pages, arXiv:0808.0227.
  16. Kitanine N., Kozlowski K.K., Maillet J.M., Slavnov N.A., Terras V., On the thermodynamic limit of form factors in the massless XXZ Heisenberg chain, J. Math. Phys. 50 (2009), 095209, 24 pages, arXiv:0903.2916.
  17. Kitanine N., Kozlowski K.K., Maillet J.M., Slavnov N.A., Terras V., Riemann-Hilbert approach to a generalised sine kernel and applications, Comm. Math. Phys. 291 (2009), 691-761, arXiv:0805.4586.
  18. Klümper A., Thermodynamics of the anisotropic spin-1/2 Heisenberg chain and related quantum chains, Z. Phys. B 91 (1993), 507-519, cond-mat/9306019.
  19. Klümper A., Batchelor M.T., An analytic treatment of finite-size corrections in the spin-1 antiferromagnetic XXZ chain, J. Phys. A: Math. Gen. 23 (1990), L189-L195.
  20. Klümper A., Wehner T., Zittartz J., Conformal spectrum of the six-vertex model, J. Phys. A: Math. Gen. 26 (1993), 2815-2827.
  21. Korepin V.E., Bogoliubov N.M., Izergin A.G., Quantum inverse scattering method and correlation functions, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 1993.
  22. Kozlowski K.K., Low-$T$ asymptotic expansion of the solution to the Yang-Yang equation, Lett. Math. Phys. 104 (2014), 55-74, arXiv:1112.6199.
  23. Kulish P.P., Reshetikhin N.Yu., Generalized Heisenberg ferromagnet and the Gross-Neveu model, Soviet Phys. JETP 80 (1981), 214-228.
  24. Lieb E.H., Wu F.Y., Absence of Mott transition in an exact solution of the short-range, one-band model in one dimension, Phys. Rev. Lett. 20 (1968), 1445-1448, Erratum, Phys. Rev. Lett. 21 (1968), 192.
  25. Orbach R., Linear antiferromagnetic chain with anisotropic coupling, Phys. Rev. 112 (1958), 309-316.
  26. Takahashi M., One-dimensional Heisenberg model at finite temperature, Progr. Theoret. Phys. 46 (1971), 401-415.
  27. Walker L.R., Antiferromagnetic linear chain, Phys. Rev. 116 (1959), 1089-1090.
  28. Yang C.N., Yang C.P., One-dimensional chain of anisotropic spin-spin interactions. II. Properties of the ground-state energy per lattice site for an infinite system, Phys. Rev. 150 (1966), 327-339.
  29. Yang C.N., Yang C.P., Thermodynamics of a one-dimensional system of bosons with repulsive delta-function interaction, J. Math. Phys. 10 (1969), 1115-1122.
  30. Zamolodchikov Al.B., Thermodynamic Bethe ansatz in relativistic models: scaling 3-state Potts and Lee-Yang models, Nuclear Phys. B 342 (1990), 695-720.

Previous article  Next article   Contents of Volume 10 (2014)