60 research outputs found
The diagonal Ising susceptibility
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
and of the corresponding susceptibility, and obtain linear
differential equations for the three and four particle contributions, as well
as the five particle contribution , 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 -particle contributions are
quite directly inherited from the direct sum structure on the form factors .
We show that the particle contributions have their
singularities at roots of unity. These singularities become dense on the unit
circle as .Comment: 18 page
High order Fuchsian equations for the square lattice Ising model:
This paper deals with , the six-particle contribution to
the magnetic susceptibility of the square lattice Ising model. We have
generated, modulo a prime, series coefficients for . The
length of the series is sufficient to produce the corresponding Fuchsian linear
differential equation (modulo a prime). We obtain the Fuchsian linear
differential equation that annihilates the "depleted" series
. The factorization of the corresponding differential
operator is performed using a method of factorization modulo a prime introduced
in a previous paper. The "depleted" differential operator is shown to have a
structure similar to the corresponding operator for . It
splits into factors of smaller orders, with the left-most factor of order six
being equivalent to the symmetric fifth power of the linear differential
operator corresponding to the elliptic integral . The right-most factor has
a direct sum structure, and using series calculated modulo several primes, all
the factors in the direct sum have been reconstructed in exact arithmetics.Comment: 23 page
Experimental mathematics on the magnetic susceptibility of the square lattice Ising model
We calculate very long low- and high-temperature series for the
susceptibility of the square lattice Ising model as well as very long
series for the five-particle contribution and six-particle
contribution . 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 10000 terms of the
series are calculated {\it modulo} a single prime, and have been used to find
the linear ODE satisfied by {\it modulo} a prime.
A diff-Pad\'e analysis of 2000 terms series for and
confirms to a very high degree of confidence previous conjectures about the
location and strength of the singularities of the -particle components of
the susceptibility, up to a small set of ``additional'' singularities. We find
the presence of singularities at for the linear ODE of ,
and for the ODE of , which are {\it not} singularities
of the ``physical'' and that is to say the
series-solutions of the ODE's which are analytic at .
Furthermore, analysis of the long series for (and )
combined with the corresponding long series for the full susceptibility
yields previously conjectured singularities in some , .
We also present a mechanism of resummation of the logarithmic singularities
of the leading to the known power-law critical behaviour occurring
in the full , and perform a power spectrum analysis giving strong
arguments in favor of the existence of a natural boundary for the full
susceptibility .Comment: 54 pages, 2 figure
Landau singularities and singularities of holonomic integrals of the Ising class
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 , with , where is the
usual Ising model coupling constant. Switching to the variable , 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 {\bf 38} (2005) 9439-9474) of the third
contribution to the magnetic susceptibility of Ising model at the
points and show that is not singular at the
corresponding points inside the unit circle , while its analytical
continuation in the variable is actually singular at the corresponding
points oustside the unit circle ().Comment: 34 pages, 1 figur
The Ising model and Special Geometries
We show that the globally nilpotent G-operators corresponding to the factors
of the linear differential operators annihilating the multifold integrals
of the magnetic susceptibility of the Ising model () are
homomorphic to their adjoint. This property of being self-adjoint up to
operator homomorphisms, is equivalent to the fact that their symmetric square,
or their exterior square, have rational solutions. The differential Galois
groups are in the special orthogonal, or symplectic, groups. This self-adjoint
(up to operator equivalence) property means that the factor operators we
already know to be Derived from Geometry, are special globally nilpotent
operators: they correspond to "Special Geometries".
Beyond the small order factor operators (occurring in the linear differential
operators associated with and ), and, in particular,
those associated with modular forms, we focus on the quite large order-twelve
and order-23 operators. We show that the order-twelve operator has an exterior
square which annihilates a rational solution. Then, its differential Galois
group is in the symplectic group . The order-23 operator
is shown to factorize in an order-two operator and an order-21 operator. The
symmetric square of this order-21 operator has a rational solution. Its
differential Galois group is, thus, in the orthogonal group
.Comment: 33 page
Difference system for Selberg correlation integrals
The Selberg correlation integrals are averages of the products
with respect to the Selberg
density. Our interest is in the case , , when this
corresponds to the -th moment of the corresponding characteristic
polynomial. We give the explicit form of a matrix linear
difference system in the variable which determines the average, and we
give the Gauss decomposition of the corresponding matrix.
For a positive integer the difference system can be used to efficiently
compute the power series defined by this average.Comment: 21 page
Holonomy of the Ising model form factors
We study the Ising model two-point diagonal correlation function 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
, the -particle contributions, . The corresponding
extension of the two-point diagonal correlation function, , is shown, for arbitrary , to be a solution of the sigma
form of the Painlev{\'e} VI equation introduced by Jimbo and Miwa. Linear
differential equations for the form factors are obtained and
shown to have both a ``Russian doll'' nesting, and a decomposition of the
differential operators as a direct sum of operators equivalent to symmetric
powers of the differential operator of the elliptic integral . Each is expressed polynomially in terms of the elliptic integrals and . The scaling limit of these differential operators breaks the
direct sum structure but not the ``Russian doll'' structure. The previous -extensions, are, for singled-out values ( integers), also solutions of linear differential
equations. These solutions of Painlev\'e VI are actually algebraic functions,
being associated with modular curves.Comment: 39 page
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