Exact expressions for row correlation functions in the isotropicd=2 ising model

Ghosh, Ranjan; Shrock, Robert

doi:10.1007/bf01010472

Cited by 20 publications

(34 citation statements)

References 7 publications

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“…The diagonal Ising two-point correlation functions can be expressed (see for instance [19,45]) as homogeneous polynomials of complete elliptic integral E and K. These diagonal Ising correlations are λ = 1 subcase of their λ-extensions C(N, N ; λ) we considered in this paper. By (7) and (8) these polynomials of E and K are also expressed as infinite sums of the form factors f (j) M,N 's which, themselves, are polynomials of E and K. This yields a double infinity (M, N ) of remarkable identities on the complete elliptic integrals E and K. Similarly, with the previous algebraic solutions for λ = cos(πm/n), one sees that an algebraic expression C(N, N ; cos(πm/n)) (associated with a modular curve) can be written as an infinite sum of polynomials in E and K. Each of these modular curves will provide a remarkable identity on the complete elliptic integrals E and K.…”

Section: Resultsmentioning

confidence: 99%

Holonomy of the Ising model form factors

Boukraa¹,

Hassani²,

Maillard³

et al. 2006

J. Phys. A: Math. Theor.

217

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Abstract. We study the Ising model two-point diagonal correlation function C(N, N ) by presenting an exponential and form factor expansion in an integral representation which differs from the known expansion of Wu, McCoy, Tracy and Barouch. We extend this expansion, weighting, by powers of a variable λ, the j-particle contributions, f N,N is expressed polynomially in terms of the complete elliptic integrals E and K. The scaling limit of these differential operators breaks the direct sum structure but not the "Russian doll" structure. The previous λ-extensions, C(N, N ; λ) are, for singledout values λ = cos(πm/n) (m, n integers), also solutions of linear differential equations. These solutions of Painlevé VI are actually algebraic functions, being associated with modular curves.

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Section: Resultsmentioning

confidence: 99%

Holonomy of the Ising model form factors

Boukraa¹,

Hassani²,

Maillard³

et al. 2006

J. Phys. A: Math. Theor.

217

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“…started long ago [9] when they were expressed in terms of determinants [10,11]. Subsequent studies of correlation functions [7,8,[12][13][14][15][16][17] led to infinite form factor expansions with every form factor given by a multiple integral [8].…”

Section: Below)mentioning

confidence: 99%

“…The normalization constant B N in (39) can be calculated by substituting expansions (16)(17) at t → 0 into (43). It appears that…”

Section: Some Facts From Painlevé VI Theorymentioning

confidence: 99%

Form factor expansions in the 2D Ising model and Painlevé VI

Mangazeev

Guttmann

2010

Nuclear Physics B

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We derive a Toda-type recurrence relation, in both high and low temperature regimes, for the λ -extended diagonal correlation functions C(N, N ; λ) of the two-dimensional Ising model, using an earlier connection between diagonal form factor expansions and tau-functions within Painlevé VI (PVI) theory, originally discovered by Jimbo and Miwa. This greatly simplifies the calculation of the diagonal correlation functions, particularly their λ-extended counterparts.We also conjecture a closed form expression for the simplest off-diagonal case C ± (0, 1; λ) where a connection to PVI theory is not known. Combined with the results for diagonal correlations these give all the initial conditions required for the λ-extended version of quadratic difference equations for the correlation functions discovered by McCoy, Perk and Wu. The results obtained here should provide a further potential algorithmic improvement in the λ-extended case, and facilitate other developments. 1

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“…Recently, the short-range correlation function X(m, n) (O~m, n~4) has exactly and explicitly been calculated by Ghosh and Shrock 19 ) on the basis of the skew-symmetric determinantal form with dimension 2m. The present expression will be advantageous to such calculations of the short-range correlation.…”

Section: )-S)mentioning

confidence: 99%