Journal of Statistical Physics

, Volume 143, Issue 1, pp 33–59 | Cite as

Ising Correlations and Elliptic Determinants

  • N. Iorgov
  • O. Lisovyy


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.


Ising model Form factor Elliptic determinant 


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© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Bogolyubov Institute for Theoretical PhysicsKyivUkraine
  2. 2.Max-Planck-Institut für MathematikBonnGermany
  3. 3.Laboratoire de Mathématiques et Physique Théorique CNRS/UMR 6083Université de Tours, Parc de GrandmontToursFrance

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