Ising model susceptibility: the Fuchsian differential equation for χ (4) and its factorization properties

J. Phys. A: Math. Theor.

et al. 2006

Self Cite

217

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.

Section: Fuchsian Linear Differential Equations Forsupporting

confidence: 65%

“…For f (2) N,N , one has: 2 f where E and K are given by (17). Other examples are given in Appendix C.…”

Section: Equivalence Of Various L J (N )'S and M J (N )'S Linear Diffmentioning

confidence: 99%

Holonomy of the Ising model form factors

Boukraa¹,

Hassani²,

J. Phys. A: Math. Theor.

et al. 2006

Self Cite

217

“…Do they disappear in χ (3) as a consequence of the enlargment of multidimensional integration domain, or as a consequence of the inclusion of the G (n) factor ? Note that, as we move closer to the original integrand and integral (3), for instance considering the multidimensional integral (87), the calculations [4,5,6] (series expansions, search of the linear ODE) become much harder. Nevertheless, an exact knowledge of the singularities of the linear ODE's can be achieved using a brand new strategy that amounts to getting very large series for these integrals modulo primes, and in a second step, get the corresponding linear ODE's also modulo primes ¶.…”

Section: Towards Ising Model Integralsmentioning

confidence: 99%

“…The linear ODE corresponding to χ (4) has [6], besides the known singularities s 2 = 1, and s 2 = −1, only the singularities (1) predicted by Nickel [1, 2], while the linear ODE corresponding to χ (3) has shown a pair of singularities 1 + 3w + 4w 2 = 0 (where w = s/2/(1 + s 2 )) that are not on the unit circle |s| = 1. The condition |s| = 1 is equivalent, for the variable w, to be real and such that its absolute value is greater than 1/4.…”

Section: Introductionmentioning

confidence: 99%

Landau singularities and singularities of holonomic integrals of the Ising class

Boukraa¹,

Hassani²,

J. Phys. A: Math. Theor.

et al. 2007

Self Cite

159

Abstract.We consider families of multiple and simple integrals of the "Ising class" and the linear ordinary differential equations with polynomial coefficients they are solutions of. We compare the full set of singularities given by the roots of the head polynomial of these linear ODE's and the subset of singularities occurring in the integrals, with the singularities obtained from the Landau conditions. For these Ising class integrals, we show that the Landau conditions can be worked out, either to give the singularities of the corresponding linear differential equation or the singularities occurring in the integral. The singular behavior of these integrals is obtained in the self-dual variable w = s/2/(1 + s 2 ), with s = sinh(2K), where K = J/kT is the usual Ising model coupling constant. Switching to the variable s, we show that the singularities of the analytic continuation of series expansions of these integrals actually break the Kramers-Wannier duality. We revisit the singular behavior (J. Phys. A 38 (2005) 9439-9474) of the third contribution to the magnetic susceptibility of Ising model χ (3) at the points 1+3w +4w 2 = 0 and show that χ (3) (s) is not singular at the corresponding points inside the unit circle |s| = 1, while its analytical continuation in the variable s is actually singular at the corresponding points 2 + s + s 2 = 0 oustside the unit circle (|s| > 1).

“…In previous studies on the Ising susceptibility [126,127,128,129], efficient programs were developed which, starting from long series expansions of a holonomic function, produce the linear ordinary differential equation (in this case Fuchsian) satisfied by the function. In order for these programs to be used to study the f N,N 's we need to efficiently produce long (up to several thousand terms) series expansions in t of the f N,N in terms of theta functions of the nome of elliptic functions, presented in [93].…”

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,

et al. 2007

SIGMA

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.