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Ising Correlations and Elliptic Determinants

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Abstract

Correlation functions of the two-dimensional Ising model on the periodic lattice can be expressed in terms of form factors—matrix elements of the spin operator in the basis of common eigenstates of the transfer matrix and translation operator. Free-fermion structure of the model implies that any multiparticle form factor is given by the pfaffian of a matrix constructed from the two-particle ones. Crossed two-particle form factors can be obtained by inverting a block of the matrix of linear transformation induced on fermions by the spin conjugation. We show that the corresponding matrix is of elliptic Cauchy type and use this observation to solve the inversion problem explicitly. Non-crossed two-particle form factors are then obtained using theta functional interpolation formulas. This gives a new simple proof of the factorized formulas for periodic Ising form factors, conjectured by A. Bugrij and one of the authors.

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References

  1. Baxter, R.J.: Superintegrable chiral Potts model: thermodynamic properties, an ‘Inverse’ model, and a simple associated Hamiltonian. J. Stat. Phys. 57, 1–39 (1989)

    Article  ADS  MathSciNet  Google Scholar 

  2. Baxter, R.J.: Some remarks on a generalization of the superintegrable chiral Potts model. J. Stat. Phys. 137, 798–813 (2009). arXiv:0906.3551 [cond-mat.stat-mech]

    Article  MATH  ADS  MathSciNet  Google Scholar 

  3. Bazhanov, V.V., Stroganov, Yu.G.: Chiral Potts model as a descendant of the six-vertex model. J. Stat. Phys. 59, 799–817 (1990)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  4. Berezin, F.A.: Method of Secondary Quantization. Nauka, Moscow (1965)

    Google Scholar 

  5. Bugrij, A.I.: Correlation function of the two-dimensional Ising model on a finite lattice: I. Theor. Math. Phys. 127, 528–548 (2001) arXiv:hep-th/0011104

    Article  MATH  Google Scholar 

  6. Bugrij, A.I., Lisovyy, O.: Spin matrix elements in 2D Ising model on the finite lattice. Phys. Lett. A 319, 390–394 (2003). arXiv:0708.3625 [nlin.SI]

    Article  MATH  ADS  MathSciNet  Google Scholar 

  7. Bugrij, A.I., Lisovyy, O.: Correlation function of the two-dimensional Ising model on a finite lattice. II. Theor. Math. Phys. 140, 987–1000 (2004). arXiv:0708.3643 [nlin.SI]

    Article  Google Scholar 

  8. Frobenius, F.G.: Über die elliptischen Funktionen zweiter Art. J. Reine Angew. Math. 93, 53–68 (1882).

    Article  Google Scholar 

  9. von Gehlen, G., Iorgov, I., Pakuliak, S., Shadura, V., Tykhyy, Yu.: Form-factors in the Baxter-Bazhanov-Stroganov model I: Norms and matrix elements. J. Phys. A 40, 14117–14138 (2007). arXiv:0708.4342 [nlin.SI]

    Article  MATH  ADS  MathSciNet  Google Scholar 

  10. von Gehlen, G., Iorgov, N., Pakuliak, S., Shadura, V., Tykhyy, Yu.: Form-factors in the Baxter-Bazhanov-Stroganov model II: Ising model on the finite lattice. J. Phys. A 41, 095003 (2008). arXiv:0711.0457 [nlin.SI]

    ADS  MathSciNet  Google Scholar 

  11. Hystad, G.: Periodic Ising correlations. arXiv:1011.2223 [math-ph]

  12. Iorgov, N., Shadura, V., Tykhyy, Yu.: Spin operator matrix elements in the quantum Ising chain: fermion approach. arXiv:1011.2603 [cond-mat.stat-mech]. To appear in J. Stat. Mech.

  13. Iorgov, N., Pakuliak, S., Shadura, V., Tykhyy, Yu., von Gehlen, G.: Spin operator matrix elements in the superintegrable chiral Potts quantum chain. J. Stat. Phys. 139, 743–768 (2010). arXiv:0912.5027 [cond-mat.stat-mech]

    Article  MATH  ADS  MathSciNet  Google Scholar 

  14. Iorgov, N.: Form-factors of the finite quantum XY-chain. arXiv:0912.4466 [cond-mat.stat-mech]

  15. Kaufman, B.: Crystal statistics. II. Partition function evaluated by spinor analysis. Phys. Rev. 76, 1232–1243 (1949)

    Article  MATH  ADS  Google Scholar 

  16. Kitanine, N., Maillet, J.M., Terras, V.: Form factors of the XXZ Heisenberg spin-1/2 finite chain. Nucl. Phys. B 554, 647–678 (1998). arXiv:math-ph/9807020

    Article  ADS  MathSciNet  Google Scholar 

  17. Korepanov, I.G.: Hidden symmetries in the 6-vertex model. Chelyabinsk Polytechnical Institute, archive VINITI No. 1472-V87, (1987)

  18. Lisovyy, O.: Transfer matrix eigenvectors of the Baxter-Bazhanov-Stroganov τ 2-model for N=2. J. Phys. A 39, 2265–2285 (2006). arXiv:nlin/0512026 [nlin.SI]

    Article  MATH  ADS  MathSciNet  Google Scholar 

  19. McCoy, B.M.: The connection between statistical mechanics and quantum field theory. In: Bazhanov, V.V., Burden, C.J. (eds.) Statistical Mechanics and Field Theory, pp. 26–128. World Scientific, Singapore (1995). arXiv:hep-th/9403084

    Google Scholar 

  20. McCoy, B.M., Wu, T.T.: The Two-Dimensional Ising Model. Harvard Univ. Press, Harvard (1973)

    MATH  Google Scholar 

  21. Nijhoff, F.W., Ragnisco, O., Kuznetsov, V.B.: Integrable time-discretization of the Ruijsenaars-Schneider model. Commun. Math. Phys. 176, 681–700 (1996). arXiv:hep-th/9412170

    Article  MATH  ADS  MathSciNet  Google Scholar 

  22. Nijhoff, F.W., Kuznetsov, V.B., Sklyanin, E.K., Ragnisco, O.: Dynamical r-matrix for the elliptic Ruijsenaars-Schneider system. J. Phys. A 29, L333–L340 (1996). arXiv:solv-int/9603006

    Article  MATH  ADS  MathSciNet  Google Scholar 

  23. Onsager, L.: Crystal statistics. I. A two-dimensional model with an order-disorder transition. Phys. Rev. 65, 117–149 (1944)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  24. Pakuliak, S., Rubtsov, V., Silantyev, A.: The SOS model partition function and the elliptic weight functions. J. Phys. A 41, 295204 (2008). arXiv:0802.0195 [math.QA]

    Article  MathSciNet  Google Scholar 

  25. Palmer, J.: Planar Ising correlations. In: Progress Mathematical Physics, vol. 49. Birkhäuser, Basel (2007)

    Google Scholar 

  26. Palmer, J., Hystad, G.: Spin matrix for the scaled periodic Ising model. J. Math. Phys. 51, 123301 (2010). arXiv:1008.0352 [nlin.SI]

    Article  ADS  MathSciNet  Google Scholar 

  27. Palmer, J., Tracy, C.A.: Two-dimensional Ising correlations: convergence of the scaling limit. Adv. Appl. Math. 2, 329–388 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  28. Rosengren, H.: An Izergin-Korepin-type identity for the 8VSOS model, with applications to alternating sign matrices. Adv. Appl. Math. 43, 137–155 (2009). arXiv:0801.1229 [math.CO]

    Article  MATH  MathSciNet  Google Scholar 

  29. Sato, M., Miwa, T., Jimbo, M.: Holonomic quantum fields V. Publ. RIMS, Kyoto Univ. 16, 531–584 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  30. Sklyanin, E.K.: Functional Bethe ansatz. In: Kupershmidt, B.A. (ed.) Integrable and Superintegrable Systems, pp. 8–33. World Scientific, Singapore (1990)

    Chapter  Google Scholar 

  31. Spiridonov, V.P.: Essays on the theory of elliptic hypergeometric functions. Russ. Math. Surv. 63, 405–472 (2008). arXiv:0805.3135 [math.CA]

    Article  MATH  MathSciNet  Google Scholar 

  32. Whittaker, E.T., Watson, G.N.: A Course of Modern Analysis, 4th edn. Cambridge Univ. Press, Cambridge (1962)

    MATH  Google Scholar 

  33. Yang, C.N.: The spontaneous magnetization of a two-dimensional Ising model. Phys. Rev. 85, 808–816 (1952)

    Article  MATH  ADS  Google Scholar 

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Iorgov, N., Lisovyy, O. Ising Correlations and Elliptic Determinants. J Stat Phys 143, 33–59 (2011). https://doi.org/10.1007/s10955-011-0154-6

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