5 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
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