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


SIGMA 7 (2011), 004, 15 pages      arXiv:1101.0035      https://doi.org/10.3842/SIGMA.2011.004
Contribution to the Proceedings of the International Workshop “Recent Advances in Quantum Integrable Systems”

Correlation Function and Simplified TBA Equations for XXZ Chain

Minoru Takahashi
Fachbereich C Physik, Bergische Universität Wuppertal, 42097 Wuppertal, Germany

Received September 27, 2010, in final form December 27, 2010; Published online January 08, 2011

Abstract
The calculation of the correlation functions of Bethe ansatz solvable models is very difficult problem. Among these solvable models spin 1/2 XXX chain has been investigated for a long time. Even for this model only the nearest neighbor and the second neighbor correlations were known. In 1990's multiple integral formula for the general correlations is derived. But the integration of this formula is also very difficult problem. Recently these integrals are decomposed to products of one dimensional integrals and at zero temperature, zero magnetic field and isotropic case, correlation functions are expressed by log 2 and Riemann's zeta functions with odd integer argument ς(3),ς(5),ς(7),.... We can calculate density sub-matrix of successive seven sites. Entanglement entropy of seven sites is calculated. These methods can be extended to XXZ chain up to n=4. Correlation functions are expressed by the generalized zeta functions. Several years ago I derived new thermodynamic Bethe ansatz equation for XXZ chain. This is quite different with Yang-Yang type TBA equations and contains only one unknown function. This equation is very useful to get the high temperature expansion. In this paper we get the analytic solution of this equation at Δ=0.

Key words: thermodynamic Bethe ansatz equation; correlation function.

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References

  1. Takahashi M., Half-filled Hubbard model at low temperature, J. Phys. C 10 (1977), 1289-1301.
  2. 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.
  3. Takahashi M., One-dimensional Heisenberg model at finite temperature, Prog. Theor. Phys. 46 (1971), 401-415.
  4. Gaudin M., Thermodynamics of the Heisenberg-Ising ring for Δ≥1, Phys. Rev. Lett. 26 (1971), 1301-1304.
  5. Takahashi M., Suzuki M., One-dimensional anisotropic Heisenberg model at finite temperatures, Prog. Theor. Phys. 46 (1972), 2187-2209.
  6. Koma T., Thermal Bethe-ansatz method for the spin-1/2 XXZ Heisenberg chain, Prog. Theor. Phys. 81 (1989), 783-809.
  7. Takahashi M., Correlation length and free energy of S=1/2 XXZ chain in magnetic field, Phys. Rev. B 44 (1991), 12382-12394.
  8. 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.
  9. Takahashi M., Simplification of thermodynamic Bethe-ansatz equations, in Physics and Combinatrix (Nagoya, 2000), World Sci. Publ., River Edge, NJ, 2001, 299-304, cond-mat/0010486.
  10. Takahashi M., Shiroishi M., Klümper A., Equivalence of TBA and QTM, J. Phys. A: Math. Gen. 34 (2001), L187-L194, cond-mat/0102027.
  11. Shiroishi M., Takahashi M., Integral equation generates high-temperature expansion of the Heisenberg chain, Phys. Rev. Lett. 89 (2002), 117201, 4 pages, cond-mat/0205180.
  12. Tsuboi Z., Takahashi M., Nonlinear integral equations for thermodynamics of the Uq(^sl(r+1)) Perk-Schultz model, J. Phys. Soc. Japan 74 (2005), 898-904, cond-mat/0412698.
  13. Hulthén L., Über das Austauschproblem eines Kristalles, Ark. Mat. Astron. Fys. A 26 (1938), 1-105.
  14. Muramoto N., Takahashi M., Integrable magnetic model of two chains coupled by four-body interactions, J. Phys. Soc. Japan 68 (1999), 2098-2104, cond-mat/9902007.
  15. Jimbo M., Miki K., Miwa T., Nakayashiki A., Correlation functions of the XXZ model for Δ<−1, Phys. Lett. A 168 (1992), 256-263, hep-th/9205055.
  16. Nakayashiki A., Some integral formulas for the solutions of the sl2 dKZ equation with level-4, Internat. J. Modern Phys. A 9 (1994), 5673-5687.
  17. Jimbo M., Miwa T., Quantum KZ equation with |q|=1 and correlation functions of the XXZ model in the gapless regime, J. Phys. A: Math. Gen. 29 (1996), 2923-2958, hep-th/9601135.
  18. Kitanine N., Maillet J.M., Terras V., Correlation functions of the XXZ Heisenberg spin-1/2 chain in a magnetic field, Nuclear Phys. B 567 (2000), 554-582, math-ph/9907019.
  19. Göhmann F., Klümper A., Seel A., Integral representations for correlation functions of the XXZ chain at finite temperature, J. Phys. A: Math. Gen. 37 (2004), 7625-7651, hep-th/0405089.
  20. Boos H.E., Korepin V.E., Quantum spin chains and Riemann zeta function with odd arguments, J. Phys. A: Math. Gen. 34 (2001), 5311-5316, hep-th/0104008.
  21. Boos H.E., Korepin V.E., Evaluation of integrals representing correlations in XXX Heisenberg spin chain, in MathPhys Odyssey (2001), Prog. Math. Phys., Vol. 23, Birkhäuser Boston, Boston, MA, 2002, 65-108, hep-th/0105144.
  22. Sakai K., Shiroishi M., Nishiyama Y., Takahashi M., Third-neighbor correlators of a one-dimensional spin-1/2 Heisenberg antiferromagnet, Phys. Rev. E 67 (2003), 065101(R), 4 pages, cond-mat/0302564.
  23. Boos H.E., Korepin V.E., Nishiyama Y., Shiroishi M., Quantum correlations and number theory, J. Phys. A: Math. Gen. 35 (2002), 4443-4451, cond-mat/0202346.
  24. Boos H.E., Korepin V.E., Smirnov F.A., Emptiness formation probability and quantum Knizhnik-Zamolodchikov equation, Nuclear Phys. B 658 (2003), 417-439, hep-th/0209246.
  25. Boos H.E., Shiroishi M., Takahashi M., First principle approach to correlation functions of spin-1/2 Heisenberg chain: fourth-neighbor correlators, Nuclear Phys. B 712 (2005), 573-599, hep-th/0410039.
  26. Sato J., Shiroishi M., Takahashi M., Correlation functions of the spin-1/2 anti-ferromagnetic Heisenberg chain: exact calculation via the generating function, Nuclear Phys. B 729 (2005), 441-466, hep-th/0507290.
  27. Boos H., Jimbo M., Miwa T., Smirnov F., Takeyama Y., A recursion formula for the correlation functions of an inhomogeneous XXX model, St. Petersburg Math. J. 17 (2005), 85-117, hep-th/0405044.
  28. Boos H., Jimbo M., Miwa T., Smirnov F., Takeyama Y., Reduced qKZ equation and correlation functions of the XXZ model, Comm. Math. Phys. 261 (2006), 245-276, hep-th/0412191.
  29. Boos H., Jimbo M., Miwa T., Smirnov F., Takeyama Y., Traces on the Sklyanin algebra and correlation functions of the eight-vertex model, J. Phys. A: Math. Gen. 38 (2005), 7629-7659, hep-th/0504072.
  30. Boos H., Jimbo M., Miwa T., Smirnov F., Takeyama Y., Density matrix of a finite sub-chain of the Heisenberg anti-ferromagnet, Lett. Math. Phys. 75 (2006), 201-208, hep-th/0506171.
  31. Kato G., Shiroishi M., Takahashi M., Sakai K., Next nearest-neighbor correlation functions of the spin-1/2 XXZ chain at critical region, J. Phys. A: Math. Gen. 36 (2003), L337-L344, cond-mat/0304475.
  32. Takahashi M., Kato G., Shiroishi M., Next nearest-neighbor correlation functions of the spin-1/2 XXZ chain at massive region, J. Phys. Soc. Japan 73 (2004), 245-253, cond-mat/0308589.
  33. Kato G., Shiroishi M., Takahashi M. Sakai K., Third-neighbour and other four-point correlation functions of spin-1/2 XXZ chain, J. Phys. A: Math. Gen. 37 (2004), 5097-5123, cond-mat/0402625.
  34. Takahashi M., Thermodynamics of one-dimensional solvable models, Cambridge University Press, Cambridge, 1999.
  35. 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.
  36. Takahashi M., On the exact ground state energy of Lieb and Wu, Prog. Theor. Phys. 45 (1971), 756-760.

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