Untitled

Abstract

We have made substantial advances in elucidating the properties of the susceptibility of the square lattice Ising model. We discuss its analyticity properties, certain closed form expressions for subsets of the coefficients, and give an algorithm of complexity O(N 6 ) to determine its first N coefficients. As a result, we have generated and analyzed series with more than 300 terms in both the high-and low-temperature regime. We quantify the effect of irrelevant variables to the scaling-amplitude functions. In particular, we find and quantify the breakdown of simple scaling, in the absence of irrelevant 1 scaling fields, arising first at order |T − T c | 9/4 , though high-low temperature symmetry is still preserved. At terms of order |T − T c | 17/4 and beyond, this symmetry is no longer present. The short-distance terms are shown to have the form (T − T c ) p (log |T − T c |) q with p ≥ q 2 . Conjectured exact expressions for some correlation functions and series coefficients in terms of elliptic theta functions also foreshadow future developments.

Holonomy of the Ising model form factors

“…We have also performed low, and high, series expansions for theĈ j (N, N ) defined by equations (4.2) in [14] (see also (20)), and we also found that these series identify with the one of h 2j (N ) and h 2j+1 (N ) with the normalization :…”

“…Recalling relations like (5.7)-(5.11) of [14], all these identities can also be written in terms of the nome of the elliptic functions occurring in the Ising model. These identities, now, become remarkable identities on some infinite Gaussian sums, or on For N = 0 this equation has been solved in terms of theta functions [40,41,42], has dihedral symmetry and has a countable number of algebraic solutions.…”

“…3, were checked against series expansions obtained independently. Some were based on extremely large series expansions, not in s or t, but in the nome of elliptic functions (see (5.7)-(5.11) of [14]), others were obtained from series expansions with hypergeometric functions coefficients. Actually our new simple integral representations (35), (36) are of a great help to produce large series expansions for the quantities f The series expansions for h 2j (N, N )(t)'s and the h 2j+1 (N, N )(t)'s agree with the series expansions for theĈ j (N, N )'s and with the series expansions for the f …”

“…The form of the result depends on whether the even, or odd, variables of [9] are integrated out. For the general anisotropic lattice, one form of this result is given, for arbitrary M and N , in [14]. When specialized to the isotropic case the result is…”

Critical behavior of one-particle spectral weights in the transverse Ising model

“…6,7 Taking the anisotropic "Hamiltonian limit" 8,9 in which the classical Ising model reduces to the quantum Ising model ͑1͒, one can show that the expressions of Wu et al reduce to Eq. ͑19͒ ͑details will be given elsewhere͒.…”