2,737 research outputs found
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
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
Singularities of -fold integrals of the Ising class and the theory of elliptic curves
We introduce some multiple integrals that are expected to have the same
singularities as the singularities of the -particle contributions
to the susceptibility of the square lattice Ising model. We find
the Fuchsian linear differential equation satisfied by these multiple integrals
for and only modulo some primes for and , thus
providing a large set of (possible) new singularities of the . We
discuss the singularity structure for these multiple integrals by solving the
Landau conditions. We find that the singularities of the associated ODEs
identify (up to ) with the leading pinch Landau singularities. The second
remarkable obtained feature is that the singularities of the ODEs associated
with the multiple integrals reduce to the singularities of the ODEs associated
with a {\em finite number of one dimensional integrals}. Among the
singularities found, we underline the fact that the quadratic polynomial
condition , that occurs in the linear differential equation
of , actually corresponds to a remarkable property of selected
elliptic curves, namely the occurrence of complex multiplication. The
interpretation of complex multiplication for elliptic curves as complex fixed
points of the selected generators of the renormalization group, namely
isogenies of elliptic curves, is sketched. Most of the other singularities
occurring in our multiple integrals are not related to complex multiplication
situations, suggesting an interpretation in terms of (motivic) mathematical
structures beyond the theory of elliptic curves.Comment: 39 pages, 7 figure
Hadronic unquenching effects in the quark propagator
We investigate hadronic unquenching effects in light quarks and mesons.
Within the non-perturbative continuum framework of Schwinger-Dyson and
Bethe-Salpeter equations we quantify the strength of the back reaction of the
pion onto the quark-gluon interaction. To this end we add a Yang-Mills part of
the interaction such that unquenched lattice results for various current quark
masses are reproduced. We find considerable effects in the quark mass function
at low momenta as well as for the chiral condensate. The quark wave function is
less affected. The Gell--Mann-Oakes-Renner relation is valid to good accuracy
up to pion masses of 400-500 MeV. As a byproduct of our investigation we verify
the Coleman theorem, that chiral symmetry cannot be broken spontaneously when
QCD is reduced to 1+1 dimensions.Comment: 27 pages, 15 figures, minor corrections and clarifications; version
to appear in PR
Exact Finite-Size-Scaling Corrections to the Critical Two-Dimensional Ising Model on a Torus
We analyze the finite-size corrections to the energy and specific heat of the
critical two-dimensional spin-1/2 Ising model on a torus. We extend the
analysis of Ferdinand and Fisher to compute the correction of order L^{-3} to
the energy and the corrections of order L^{-2} and L^{-3} to the specific heat.
We also obtain general results on the form of the finite-size corrections to
these quantities: only integer powers of L^{-1} occur, unmodified by logarithms
(except of course for the leading term in the specific heat); and the
energy expansion contains only odd powers of L^{-1}. In the specific-heat
expansion any power of L^{-1} can appear, but the coefficients of the odd
powers are proportional to the corresponding coefficients of the energy
expansion.Comment: 26 pages (LaTeX). Self-unpacking file containing the tex file and
three macros (indent.sty, eqsection.sty, subeqnarray.sty). Added discussions
on the results and new references. Version to be published in J. Phys.
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
Surface Photovoltage Spectroscopy over Wide Time Domains for Semiconductors with Ultrawide Bandgap Example of Gallium Oxide
A nonconventional approach is proposed for the measurement of surface photovoltage SPV signals over very wide ranges in photon energy and time. Regimes for AC, DC, and combined AC DC measurements are defined and applied for the characterization of a amp; 946; Ga2O3 crystal by transient and modulated SPV spectroscopy, spectroscopy in the mode of a Kelvin probe and single pulse SPV transients from 10 amp; 8201;ns to 1000 amp; 8201;s with the same electrode. Numerous electronic transitions are distinguished in amp; 946; Ga2O3 depending on the measurement regime, the time response and the history of measurement. An accumulation of negative charge at the surface of amp; 946; Ga2O3 is observed at long times independent whether the SPV signals are positive or negative before. The nonconventional approach of measuring SPV signals opens new opportunities for investigating electronic properties of semiconductors with ultrawide bandgaps and any other photoactive material
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
Renormalization Group calculations with k|| dependent couplings in a ladder
We calculate the phase diagram of a ladder system, with a Hubbard interaction
and an interchain coupling . We use a Renormalization Group method, in
a one loop expansion, introducing an original method to include
dependence of couplings. We also classify the order parameters corresponding to
ladder instabilities. We obtain different results, depending on whether we
include dependence or not. When we do so, we observe a region with
large antiferromagnetic fluctuations, in the vicinity of small ,
followed by a superconducting region with a simultaneous divergence of the Spin
Density Waves channel. We also investigate the effect of a non local backward
interchain scattering : we observe, on one hand, the suppression of singlet
superconductivity and of Spin Density Waves, and, on the other hand, the
increase of Charge Density Waves and, for some values of , of triplet
superconductivity. Our results eventually show that is an influential
variable in the Renormalization Group flow, for this kind of systems.Comment: 20 pages, 19 figures, accepted in Phys. Rev. B 71 v. 2
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