Addendum to `On the singularity structure of the 2D Ising model susceptibility'

Nickel, B. G.

doi:10.1088/0305-4470/33/8/313

Cited by 58 publications

(331 citation statements)

References 24 publications

Supporting

Mentioning

308

Contrasting

Unclassified

Order By: Relevance

“…All the previous results confirm that these various λ extensions identify and actually verify (11). The (log-derivative) of the λ extensions C(N, N ; λ) satisfy the same (sigma-form of ) Painlevé VI equation (11) as the original diagonal spin-spin correlation, the boundary condition dependence coming from the original diagonal spin-spin correlation boundary condition.…”

Section: Series Solution Of the Sigma Form Of Painlevé VIsupporting

confidence: 72%

“…(2). The normalization condition (13) fixes one integration constant in the solution to (11). We find that the second integration constant is a free parameter, and, denoting that parameter by λ, that our one parameter family of solutions for C − (N, N ) can be written in a form structurally similar to the right hand side of (9).…”

Section: Introductionmentioning

confidence: 85%

“…For T < T c one such set♯ of parameters of (102), corresponding to the N -dependent sigma form (11), is :…”

Section: Algebraic Solutions Of Pvi For λ = Cos(πm/n) and Modular Curvesmentioning

confidence: 99%

“…On another hand, Jimbo and Miwa introduced in [20] an isomonodromic λ-extension of C(N, N ) and showed that this more general function C(N, N ; λ) also satisfies (11). The motivation of introducing an isomonodromic parameter λ, in the framework of isomonodromy deformations, is, at first sight, quite different from the "coupling constant" motivation at the origin of the form factor λ-extensions (9) and (10).…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Holonomy of the Ising model form factors

Boukraa¹,

Hassani²,

Maillard³

et al. 2006

J. Phys. A: Math. Theor.

217

View full text Add to dashboard Cite

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.

show abstract

Section: Series Solution Of the Sigma Form Of Painlevé VIsupporting

confidence: 72%

Section: Introductionmentioning

confidence: 85%

“…For T < T c one such set♯ of parameters of (102), corresponding to the N -dependent sigma form (11), is :…”

Section: Algebraic Solutions Of Pvi For λ = Cos(πm/n) and Modular Curvesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Holonomy of the Ising model form factors

Boukraa¹,

Hassani²,

Maillard³

et al. 2006

J. Phys. A: Math. Theor.

217

View full text Add to dashboard Cite

show abstract

“…The form factor expressions for the two-point correlation functions C(M, N ) of [87,88,94,93,123,124] are obtained by expanding the exponentials in (1.1), and (1.2), in the form given in [123] as multiple integrals, and integrating over half the variables. The form of the result depends on whether the even, or odd, variables of [123] are integrated out.…”

Section: Nn 'Smentioning

confidence: 99%

From Holonomy of the Ising Model Form Factors to n-Fold Integrals and the Theory of Elliptic Curves

Boukraa

Hassani²,

Maillard³

et al. 2007

SIGMA

View full text Add to dashboard Cite

Abstract. We recall the form factors f (j) N,N corresponding to the λ-extension C(N, N ; λ) of the two-point diagonal correlation function of the Ising model on the square lattice and their associated linear differential equations which exhibit both a "Russian-doll" nesting, and a decomposition of the linear differential operators as a direct sum of operators (equivalent to symmetric powers of the differential operator of the complete elliptic integral E). The scaling limit of these differential operators breaks the direct sum structure but not the "Russian doll" structure, the "scaled" linear differential operators being no longer Fuchsian. We then introduce some multiple integrals of the Ising class expected to have the same singularities as the singularities of the n-particle contributions χ (n) to the susceptibility of the square lattice Ising model. We find the Fuchsian linear differential equations satisfied by these multiple integrals for n = 1, 2, 3, 4 and, only modulo a prime, for n = 5 and 6, thus providing a large set of (possible) new singularities of the χ (n) . We get the location of these singularities by solving the Landau conditions. We discuss the mathematical, as well as physical, interpretation of these new singularities. Among the singularities found, we underline the fact that the quadratic polynomial condition 1 + 3w + 4w 2 = 0, that occurs in the linear differential equation of χ (3) , actually corresponds to the occurrence of complex multiplication for elliptic curves. The interpretation of complex multiplication for elliptic curves as complex fixed points of generators of the exact renormalization group is sketched. The other singularities occurring in our multiple integrals are not related to complex multiplication situations, suggesting a geometric interpretation in terms of more general (motivic) mathematical structures beyond the theory of elliptic curves. The scaling limit of the (lattice off-critical) structures as a confluent limit of regular singularities is discussed in the conclusion.

show abstract