Journal of Mathematical Sciences

, Volume 216, Issue 1, pp 8–22 | Cite as

Combinatorial Aspects of Correlation Functions of the XXZ Heisenberg Chain in Limiting Cases

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We discuss the connection between quantum integrable models and some aspects of enumerative combinatorics and the theory of partitions. As a basic example, we consider the spin XXZ Heisenberg chain in the limiting cases of zero and infinite anisotropy. The representation of the Bethe wave functions via Schur functions allows us to apply the theory of symmetric functions to calculating the thermal correlation functions as well as the form factors in the determinantal form. We provide a combinatorial interpretation of the correlation functions in terms of nests of self-avoiding lattice paths. The suggested interpretation is in turn related to the enumeration of boxed plane partitions. The asymptotic behavior of the thermal correlation functions is studied in the limit of small temperature provided that the characteristic parameters of the system are large enough. The leading asymptotics of the correlators are found to be proportional to the squared numbers of boxed plane partitions.

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References

  1. 1.
    G. E. Andrews, The Theory of Partitions, Cambridge Univ. Press, Cambridge (1998).MATHGoogle Scholar
  2. 2.
    E. W. Barnes, “The theory of the G-function,” Quart. J. Pure Appl. Math., 31, 264–314 (1900).MATHGoogle Scholar
  3. 3.
    N. M. Bogoliubov, “XX Heisenberg chain and random walks,” J. Math. Sci., 138, No. 3, 5636–5643 (2006).MathSciNetCrossRefGoogle Scholar
  4. 4.
    N. M. Bogoliubov, “The integrable models for the vicious and friendly walkers,” J. Math. Sci., 143, No. 1, 2729–2737 (2007).MathSciNetCrossRefGoogle Scholar
  5. 5.
    N. M. Bogoliubov, A. G. Izergin, and V. E. Korepin, Correlation Functions of Integrable Systems and Quantum Inverse Scattering Method [in Russian], Nauka, Moscow (1992).MATHGoogle Scholar
  6. 6.
    N. M. Bogoliubov and C. Malyshev, “The correlation functions of the XX Heisenberg magnet and random walks of vicious walkers,” Theor. Math. Phys., 159, No. 2, 179–192 (2009).MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    N. M. Bogoliubov and C. Malyshev, “The correlation functions of the XXZ Heisenberg chain in the case of zero or infinite anisotropy, and random walks of vicious walkers,” St.Petersburg Math. J., 22, No. 3, 359–377 (2011).MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    N. M. Bogoliubov and C. L. Malyshev, “The Ising limit of the XXZ Heisenberg magnet and certain thermal correlation functions,” Theor. Math. Phys., 169, No. 2, 1517–1529 (2011).MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    N. M. Bogoliubov and C. Malyshev, “Correlation functions of XX0 Heisenberg chain, q-binomial determinants, and random walks,” Nucl. Phys. B, 879, 268–291 (2014).MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    N. M. Bogoliubov and C. L. Malyshev, “A combinatorial interpretation of the scalar products of state vectors of integrable models,” Zap. Nauchn. Semin. POMI, 421, 35–45 (2014).MATHGoogle Scholar
  11. 11.
    A. Borodin, V. Gorin, and E. M. Rains, “q-Distributions on boxed plane partitions,” Selecta Math. (N. S.), 16, No. 4, 731–789 (2010).MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    A. Borodin and G. Olshanski, “Infinite-dimensional diffusions as limits of random walks on partitions,” Prob. Theory Related Fields, 144, No. 1–2, 281–318 (2009).MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    D. M. Bressoud, Proofs and Confirmations. The Story of the Alternating Sign Matrix Conjecture, Cambridge Univ. Press, Cambridge (1999).CrossRefMATHGoogle Scholar
  14. 14.
    L. D. Faddeev and L. A. Takhtajan, “Quantum inverse scattering method and the XYZ Heisenberg model,” Uspekhi Mat. Nauk, 34, No. 5(209), 13–63 (1979).MathSciNetGoogle Scholar
  15. 15.
    P. J. Forrester, Log-Gases and Random Matrices, Princeton Univ. Press, Princeton (2010).MATHGoogle Scholar
  16. 16.
    I. Gessel and G. Viennot, “Binomial determinants, paths, and hook length formulae,” Adv. in Math., 58, No. 3, 300–321 (1985).MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    V. E. Korepin, N. M. Bogoliubov, and A. G. Izergin, Quantum Inverse Scattering Method and Correlation Functions, Cambridge Univ. Press, Cambridge (1993).CrossRefMATHGoogle Scholar
  18. 18.
    C. Krattenthaler, A. J. Guttmann, and X. G. Viennot, “Vicious walkers, friendly walkers and Young tableaux: II. With a wall,” J. Phys. A, 33, No. 48, 8835–8866 (2000).MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    G. Kuperberg, “Another proof of the alternating-sign matix conjecture,” Int. Math. Res. Notices, 1996, 139–150 (1996).MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    I. G. Macdonald, Symmetric Functions and Hall Polynomials, Oxford Univ. Press, Oxford (1995).MATHGoogle Scholar
  21. 21.
    P. A. MacMahon, Combinatory Analysis, Vols. 1, 2. Cambridge Univ. Press, Cambridge (1915), (1916).Google Scholar
  22. 22.
    S. N. Majumdar and G. Schehr, “Top eigenvalue of a random matrix: large deviations and third order phase transition,” J. Stat. Mech., 2014, P01012 (2014).MathSciNetCrossRefGoogle Scholar
  23. 23.
    W. H. Mills, D. P. Robbins, and H. Rumsey, Jr., “Alternating sign matrices and descending plane partitions,” J. Combin. Theory Ser. A, 34, No. 3, 340–359 (1983).MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    A. Okounkov and N. Reshetikhin, “Correlation function of Schur process with application to local geometry of a random 3-dimensional Young diagram,” J. Amer. Math. Soc., 16, No. 3, 581–603 (2003).MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    A. Okounkov, N. Reshetikhin, and C. Vafa, “Quantum Calabi–Yau and classical crystals,” in: P. Etingof, V. S. Retakh, and I. M. Singer (eds.), The Unity of Mathematics (In Honor of the Ninetieth Birthday of I. M. Gelfand), Birkhäuser, Boston (2006), pp. 597–618.Google Scholar
  26. 26.
    D. Pérez-Garcia and M. Tierz, “Mapping between the Heisenberg XX spin chain and low-energy QCD,” Phys. Rev. X, 4, 021050 (2003).Google Scholar
  27. 27.
    R. Stanley, Enumerative Combinatorics, Vol. 2. Cambridge Univ. Press, Cambridge (1999).CrossRefMATHGoogle Scholar
  28. 28.
    N. Tsilevich, “Quantum inverse scattering method for the q-boson model and symmetric functions,” Funct. Anal. Appl., 40, No. 3, 207–217 (2006).MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    A. Vershik, “Statistical mechanics of combinatorial partitions, and their limit configurations,” Funct. Anal. Appl., 30, No. 2, 90–105 (1996).MathSciNetCrossRefMATHGoogle Scholar

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© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.St.Petersburg Department of Steklov Mathematical InstituteITMO UniversitySt.PetersburgRussia
  2. 2.St.Petersburg Department of Steklov Mathematical InstituteSt.PetersburgRussia

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