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Register a new userThe saga of the Ising susceptibility
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We review developments made since 1959 in the search for a closed form for the susceptibility of the Ising model. The expressions for the form factors in terms of the nome $q$ and the modulus $k$ are compared and contrasted. The $lambda$ generalized correlations $C(M,N;lambda)$ are defined and explicitly computed in terms of theta functions for $M=N=0,1$.
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We study the full susceptibility of the Ising model modulo powers of primes. We find exact functional equations for the full susceptibility modulo these primes. Revisiting some lesser-known results on discrete finite automata, we show that these results can be seen as a consequence of the fact that, modulo 2^r, one cannot distinguish the full susceptibility from some simple diagonals of rational functions which reduce to algebraic functions modulo 2^r, and, consequently, satisfy exact functional equations modulo 2^r. We sketch a possible physical interpretation of these functional equations modulo 2^r as reductions of a master functional equation corresponding to infinite order symmetries such as the isogenies of elliptic curves. One relevant example is the Landen transformation which can be seen as an exact generator of the Ising model renormalization group. We underline the importance of studying a new class of functions corresponding to ratios of diagonals of rational functions: they reduce to algebraic functions modulo powers of primes and they may have solutions with natural boundaries.
We use the recently derived form factor expansions of the diagonal two-point correlation function of the square Ising model to study the susceptibility for a magnetic field applied only to one diagonal of the lattice, for the isotropic Ising model. We exactly evaluate the one and two particle contributions $chi_{d}^{(1)}$ and $chi_{d}^{(2)}$ of the corresponding susceptibility, and obtain linear differential equations for the three and four particle contributions, as well as the five particle contribution ${chi}^{(5)}_d(t)$, but only modulo a given prime. We use these exact linear differential equations to show that, not only the russian-doll structure, but also the direct sum structure on the linear differential operators for the $ n$-particle contributions $chi_{d}^{(n)}$ are quite directly inherited from the direct sum structure on the form factors $ f^{(n)}$. We show that the $ n^{th}$ particle contributions $chi_{d}^{(n)}$ have their singularities at roots of unity. These singularities become dense on the unit circle $|sinh2E_v/kT sinh 2E_h/kT|=1$ as $ nto infty$.
We give the exact expressions of the partial susceptibilities $chi^{(3)}_d$ and $chi^{(4)}_d$ for the diagonal susceptibility of the Ising model in terms of modular forms and Calabi-Yau ODEs, and more specifically, $_3F_2([1/3,2/3,3/2],, [1,1];, z)$ and $_4F_3([1/2,1/2,1/2,1/2],, [1,1,1]; , z)$ hypergeometric functions. By solving the connection problems we analytically compute the behavior at all finite singular points for $chi^{(3)}_d$ and $chi^{(4)}_d$. We also give new results for $chi^{(5)}_d$. We see in particular, the emergence of a remarkable order-six operator, which is such that its symmetric square has a rational solution. These new exact results indicate that the linear differential operators occurring in the $n$-fold integrals of the Ising model are not only Derived from Geometry (globally nilpotent), but actually correspond to Special Geometry (homomorphic to their formal adjoint). This raises the question of seeing if these special geometry Ising-operators, are special ones, reducing, in fact systematically, to (selected, k-balanced, ...) $_{q+1}F_q$ hypergeometric functions, or correspond to the more general solutions of Calabi-Yau equations.
We calculate very long low- and high-temperature series for the susceptibility $chi$ of the square lattice Ising model as well as very long series for the five-particle contribution $chi^{(5)}$ and six-particle contribution $chi^{(6)}$. These calculations have been made possible by the use of highly optimized polynomial time modular algorithms and a total of more than 150000 CPU hours on computer clusters. For $chi^{(5)}$ 10000 terms of the series are calculated {it modulo} a single prime, and have been used to find the linear ODE satisfied by $chi^{(5)}$ {it modulo} a prime. A diff-Pade analysis of 2000 terms series for $chi^{(5)}$ and $chi^{(6)}$ confirms to a very high degree of confidence previous conjectures about the location and strength of the singularities of the $n$-particle components of the susceptibility, up to a small set of ``additional singularities. We find the presence of singularities at $w=1/2$ for the linear ODE of $chi^{(5)}$, and $w^2= 1/8$ for the ODE of $chi^{(6)}$, which are {it not} singularities of the ``physical $chi^{(5)}$ and $chi^{(6)},$ that is to say the series-solutions of the ODEs which are analytic at $w =0$. Furthermore, analysis of the long series for $chi^{(5)}$ (and $chi^{(6)}$) combined with the corresponding long series for the full susceptibility $chi$ yields previously conjectured singularities in some $chi^{(n)}$, $n ge 7$. We also present a mechanism of resummation of the logarithmic singularities of the $chi^{(n)}$ leading to the known power-law critical behaviour occurring in the full $chi$, and perform a power spectrum analysis giving strong arguments in favor of the existence of a natural boundary for the full susceptibility $chi$.
We study the class of non-holonomic power series with integer coefficients that reduce, modulo primes, or powers of primes, to algebraic functions. In particular we try to determine whether the susceptibility of the square-lattice Ising model belongs to this class, and more broadly whether the susceptibility is a solution of a differentially algebraic equation. Initial results on Tuttes non-linear ordinary differential equation (ODE) and other simple quadratic non-linear ODEs suggest that a large set of differentially algebraic power series solutions with integer coefficients might reduce to algebraic functions modulo primes, or powers of primes. Here we give several examples of series with integer coefficients and non-zero radius of convergence that reduce to algebraic functions modulo (almost) every prime (or power of a prime). These examples satisfy differentially algebraic equations with the encoding polynomial occasionally possessing quite high degree (and thus difficult to identify even with long series). Additionally, we have extended both the high- and low-temperature Ising square-lattice susceptibility series to 5043 coefficients. We find that even this long series is insufficient to determine whether it reduces to algebraic functions modulo $3$, $5$, etc. This negative result is in contrast to the comparatively easy confirmation that the corresponding series reduce to algebraic functions modulo powers of $2$.
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