We show that the n-fold integrals $\chi^{(n)}$ of the magnetic susceptibility of the Ising model, as well as various other n-fold integrals of the "Ising class", or n-fold integrals from enumerative combinatorics, like lattice Green functions, are actually diagonals of rational functions. As a consequence, the power series expansions of these solutions of linear differential equations "Derived From Geometry" are globally bounded, which means that, after just one rescaling of the expansion variable, they can be cast into series expansions with integer coefficients. Besides, in a more enumerative combinatorics context, we show that generating functions whose coefficients are expressed in terms of nested sums of products of binomial terms can also be shown to be diagonals of rational functions. We give a large set of results illustrating the fact that the unique analytical solution of Calabi-Yau ODEs, and more generally of MUM ODEs, is, almost always, diagonal of rational functions. We revisit Christol's conjecture that globally bounded series of G-operators are necessarily diagonals of rational functions. We provide a large set of examples of globally bounded series, or series with integer coefficients, associated with modular forms, or Hadamard product of modular forms, or associated with Calabi-Yau ODEs, underlying the concept of modularity. We finally address the question of the relations between the notion of integrality (series with integer coefficients, or, more generally, globally bounded series) and the modularity (in particular integrality of the Taylor coefficients of mirror map), introducing new representations of Yukawa couplings.

Ising n-fold integrals as diagonals of rational functions and integrality of series expansions: integrality versus modularity

The first experimental study of the statistics of a quasi-one-dimensional Ising system under the magnetic field H α , described by the Hamiltonian \begin{aligned} \mathscr{H}=-2J_{1} \sum _{\langle ij \rangle} S_{i}^{z}S_{j}^{z}-2J_{2} \sum _{\langle ik \rangle} S_{i}^{z}S_{k}^{z}-g_{\alpha}\mu _{\text{B}}H^{\alpha} \sum _{i} S_{i}^{\alpha}\hspace{1em}(\alpha=x,~y,~z) \label{e1} \end{aligned} has been performed, where J 1 and J 2 are the intra- and the inter-chain exchange constants, respectively. A single crystal of the compound (CH 3 ) 3 NHCoCl 3 ·2H 2 O has been used as a model sample of the ferromagnetic system with J 1 / k B =14.2 K and J 2 / k B =0.20 K. It has been revealed that the experimental values of the magnetic heat capacity under the field H α >2 J 2 / g z µ B (≈0.8 kOe) applied along the spin preferential axis are excellently reproduced by the values calculated for the isolated Ising chain under the longitudinal field ( α= z ; g z =6.54). For the temperature higher than 7 K (≈ J 1 /2 k B )...

An Experimental Study of the Ising Chain Statistics under the Magnetic Field

For the analysis of finite-dimensional classical completely integrable Hamiltonian systems the representation by action-angle-variables is an essential tool. This representation can be carried over to the infinite-dimensional situation by use of mastersymmetries, thus leading to a suitable Viasoro algebra in the vector fields. For the quantum case, however, such a structure cannot exist in the corresponding operator algebra, due to a classical theorem of Kaplansky, although the concept of mastersymmetries can be used to give a formal description of the infinitedimensional symmetry group of those “nonlinear” Quantum systems which are accessible by the Quantum Inverse Scattering Transform. In order to make that precise and to transfer classical notions and methods to the quantum case an alternative dynamical concept for quantum systems is proposed. We give two examples, in the discrete case we consider spin chains, like the Heisenberg anisotropic spin chain, and in the continuous case, where additional difficulties arise, we consider the quantization of the KdV.

/pdf/an-alternative-dynamical-description-of-quantum-systems-p2hfwnzsmt.pdf

An alternative dynamical description of Quantum Systems

On exprime la fonction de partition du modele d'Ising a 2 dimensions en presence d'un champ magnetique par une integrale quadruple d'un produit double de fonctions elementaires

A Two‐Dimensional Ising Model in a Magnetic Field—A Scalar Representation of the Partition Function

We show that the n-fold integrals χ(n) of the magnetic susceptibility of the Ising model, as well as various other n-fold integrals of the ‘Ising class’, or n-fold integrals from enumerative combinatorics, like lattice Green functions, correspond to a distinguished class of functions generalizing algebraic functions: they are actually diagonals of rational functions. As a consequence, the power series expansions of the, analytic at x = 0, solutions of these linear differential equations ‘derived from geometry’ are globally bounded, which means that after just one rescaling of the expansion variable, they can be cast into series expansions with integer coefficients. We also give several results showing that the unique analytical solution of Calabi–Yau ODEs and, more generally, Picard–Fuchs linear ODEs with solutions of maximal weights are always diagonals of rational functions. Besides, in a more enumerative combinatorics context, generating functions whose coefficients are expressed in terms of nested sums of products of binomial terms can also be shown to be diagonals of rational functions. We finally address the question of the relations between the notion of integrality (series with integer coefficients, or, more generally, globally bounded series) and the modularity of ODEs.

Ising n-fold integrals as diagonals of rational functions and integrality of series expansions

On the Ising model in the presence of magnetic field

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On the Ising model in the presence of magnetic field