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St. Petersburg Mathematical Journal

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A recursion formula for the correlation functions of an inhomogeneous XXX model

Authors: H. Boos, M. Jimbo, T. Miwa, F. Smirnov and Y. Takeyama
Original publication: Algebra i Analiz, tom 17 (2005), nomer 1.
Journal: St. Petersburg Math. J. 17 (2006), 85-117
MSC (2000): Primary 82B23
Published electronically: January 23, 2006
MathSciNet review: 2140676
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Abstract | References | Similar Articles | Additional Information

Abstract: A new recursion formula is presented for the correlation functions of the integrable spin $ 1/2$ XXX chain with inhomogeneity. It links the correlators involving $ n$ consecutive lattice sites to those with $ n-1$ and $ n-2$ sites. In a series of papers by V. Korepin and two of the present authors, it was discovered that the correlators have a certain specific structure as functions of the inhomogeneity parameters. The formula mentioned above makes it possible to prove this structure directly, as well as to obtain an exact description of the rational functions that were left undetermined in the earlier work.

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  • 1. R. Baxter, Exactly solved models in statistical mechanics, Acad. Press, Inc., London, 1982. MR 0690578 (86i:82002a)
  • 2. V. Bazhanov, S. Lukyanov, and A. Zamolodchikov, Integrable structure of conformal field theory. II. $ Q$-operator and DDV equation, Comm. Math. Phys. 190 (1997), 247-278. MR 1489571 (99h:81191)
  • 3. H. Bethe, Zur Theorie der Metalle. I. Eigenwerte und Eigenfunktionen der linearen Atomkette, Z. Phys. 71 (1931), 205.
  • 4. H. Boos and V. Korepin, Quantum spin chains and Riemann zeta functions with odd arguments, hep-th/0104008; J. Phys. A 34 (2001), 5311-5316. MR 1855758 (2002i:82009)
  • 5. H. Boos, V. Korepin, and F. Smirnov, Emptiness formation probability and quantum Knizhnik-Zamolodchikov equation, hep-th/0209246; Nuclear Phys. B 658 (2003), no. 3, 417-439. MR 1976325 (2004m:82028)
  • 6. -, New formulae for solutions of quantum Knizhnik-Zamolodchikov equations on level $ -4$, hep-th/0304077; J. Phys. A 37 (2004), 323-335. MR 2046886 (2005e:82024)
  • 7. -, New formulae for solutions of quantum Knizhnik-Zamolodchikov equations on level $ -4$ and correlation functions, hep-th/0305135.
  • 8. H. Boos, M. Shiroishi, and M. Takahashi, First principle approach to correlation functions of spin-$ 1/2$ Heisenberg chain: fourth-neighbor correlators (work in progress).
  • 9. L. Faddeev, How algebraic Bethe ansatz works for integrable models, Symétries Quantiques (Les Houches, 1995) (A. Connes, K. Gawedzki, J. Zinn-Justin, eds.), North-Holland, Amsterdam, 1998, pp. 149-219. MR 1616371 (2000b:82010)
  • 10. L. Faddeev and L. Takhtajan, The spectrum and scattering of excitations in the one-dimensional isotropic Heisenberg model, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 109 (1981), 134-178; English transl., J. Soviet Math. 24 (1984), no. 2, 241-267. MR 0629119 (83b:82022)
  • 11. I. Frenkel and N. Reshetikhin, Quantum affine algebras and holonomic difference equations, Comm. Math. Phys. 146 (1992), 1-60. MR 1163666 (94c:17024)
  • 12. M. Jimbo and T. Miwa, Algebraic analysis of solvable lattice models, CBMS Regional Conf. Ser. in Math., vol. 85, Amer. Math. Soc., Providence, RI, 1995. MR 1308712 (96e:82037)
  • 13. M. Jimbo, T. Miwa, K. Miki, and A. Nakayashiki, Correlation functions of the XXZ model for $ \Delta<-1$, Phys. Lett. A 168 (1992), 256-263. MR 1178036 (93m:82007)
  • 14. G. Kato, M. Shiroishi, M. Takahashi, and K. Sakai, Third-neighbor and other four-point correlation functions of spin-1/2 XXZ chain, J. Phys. A 37 (2004), 5097-5123. MR 2066956 (2005f:82017)
  • 15. -, Next-nearest-neighbor correlation functions of the spin-1/2 XXZ chain at the critical region, J. Phys. A 36 (2003), L337-L344. MR 1986946
  • 16. N. Kitanine, J.-M. Maillet, and V. Terras, Correlation functions of the XXZ Heisenberg spin- $ \frac{1}{2}$-chain in a magnetic field, Nuclear Phys. B 567 (2000), 554-582. MR 1741654 (2001k:82022)
  • 17. V. Korepin, A. Izergin, F. Essler, and D. Uglov, Correlation function of the spin-1/2 XXX antiferromagnet, cond-mat/9403066; Phys. Lett. A 190 (1994), 182-184. MR 1283785 (95f:82013)
  • 18. P. Kulish, N. Reshetikhin, and E. Sklyanin, Yang-Baxter equations and representation theory. I, Lett. Math. Phys. 5 (1981), 393-403. MR 0649704 (83g:81099)
  • 19. K. Sakai, M. Shiroishi, Y. Nishiyama, and M. Takahashi, Third neighbor correlators of a one-dimensional spin-1/2 Heisenberg antiferromagnet, Phys. Rev. E 67 (2003), 065101.
  • 20. F. Smirnov, Form factors in completely integrable models of quantum field theory, Adv. Ser. Math. Phys., vol. 14, World Sci. Publ. Co., Inc., River Edge, NJ, 1992. MR 1253319 (95a:81254)
  • 21. -, Dynamical symmetries of massive integrable models, Infinite Analysis, Part A, B (Kyoto, 1991), Adv. Ser. Math. Phys., vol. 16, World Sci. Publ., River Edge, NJ, 1992, pp. 813-858. MR 1187577 (94b:81118); MR 1187578 (94b:81119)
  • 22. M. Takahashi, Half-filled Hubbard model at low temperature, J. Phys. C 10 (1977), 1298.
  • 23. M. Takahashi, G. Kato, and M. Shiroishi, Next nearest-neighbor correlation functions of the spin-1/2 XXZ chain at massive region, J. Phys. Soc. Japan. 73 (2004), 245.

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Additional Information

H. Boos
Affiliation: Physics Department, University of Wuppertal, D-42097, Wuppertal, Germany, and Institute for High Energy Physics, Protvino 142284, Russia

M. Jimbo
Affiliation: Graduate School of Mathematical Sciences, University of Tokyo, Tokyo 153-8914, Japan

T. Miwa
Affiliation: Department of Mathematics, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan

F. Smirnov
Affiliation: (Membre du CNRS): Laboratoire de Physique Théorique et Hautes Energies, Université Pierre et Marie Curie, Tour 16 1$^{er}$ étage, 4 Place Jussieu, 75252 Paris Cedex 05, France

Y. Takeyama
Affiliation: Graduate School of Pure and Applied Sciences, University of Tsukuba, Tsukuba 305-8571, Japan

Keywords: Exactly solvable models, spin chains, correlation functions
Received by editor(s): October 9, 2004
Published electronically: January 23, 2006
Additional Notes: Research of H. Boos was supported by INTAS grant #00–00561 and by RFBR grant #04–01–00352.
Research of M. Jimbo was partially supported by the Grant-in-Aid for Scientific Research B2–16340033.
Research of T. Miwa was partially supported by the Grant-in-Aid for Scientific Research A1–13304010.
Research of F. Smirnov was supported by INTAS grant #00–00055 and by the EC network “EUCLID”, contract number HPRN–CT–2002–00325.
Research of Y. Takeyama was partially supported by the University of Tsukuba Research Project.
This work was started during the workshop, 21COE RIMS Research Project 2004, Quantum Integrable Systems and Infinite-Dimensional Algebras, February 4–24, 2004.
Dedicated: Dedicated to Ludwig Faddeev on the occasion of his seventieth birthday
Article copyright: © Copyright 2006 American Mathematical Society

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