Experimental mathematics on the magnetic susceptibility of the square lattice Ising model
Identifieur interne : 009658 ( Main/Merge ); précédent : 009657; suivant : 009659Experimental mathematics on the magnetic susceptibility of the square lattice Ising model
Auteurs : S. Boukraa [Algérie] ; A. J. Guttmann [Australie] ; S. Hassani [Algérie] ; I. Jensen [Australie] ; J-M. Maillard [France] ; B. Nickel [Canada] ; N. Zenine [Algérie]Source :
- Journal of physics. A, Mathematical and theoretical : (Print) [ 1751-8113 ] ; 2008.
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- Pascal (Inist)
English descriptors
- KwdEn :
Abstract
We calculate very long low- and high-temperature series for the susceptibility X of the square lattice Ising model as well as very long series for the five-particle contribution X(5) and six-particle contribution X(6). These calculations have been made possible by the use of highly optimized polynomial time modular algorithms and a total of more than 150 000 CPU hours on computer clusters. The series for / (low- and high-temperature regimes), X(5) and X(6) are now extended to 2000 terms. In addition, for X(5), 10000 terms of the series are calculated modulo a single prime, and have been used to find the linear ODE satisfied by X(5) modulo a prime. A diff-Padé analysis of the 2000 terms series for X(5) and X(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. The exponents at all the singularities of the Fuchsian linear ODE of X(5) and the (as yet unknown) ODE of X(6) are given: they are all rational numbers. We find the presence of singularities at w = 1/2 for the linear ODE of X(5), and w2 = 1/8 for the ODE of X(6), which are not singularities of the 'physical' X(5) and X(6), that is to say the series solutions of the ODE's which are analytic at w = 0. Furthermore, analysis of the long series for X(5) (and X(6)) combined with the corresponding long series for the full susceptibility X yields previously conjectured singularities in some X(n), n ≥ 7. The exponents at all these singularities are also seen to be rational numbers. We also present a mechanism of resummation of the logarithmic singularities of the X(n) leading to the known power-law critical behaviour occurring in the full X, and perform a power spectrum analysis giving strong arguments in favour of the existence of a natural boundary for the full susceptibility X.
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<profileDesc><textClass><keywords scheme="KwdEn" xml:lang="en"><term>Algorithms</term>
<term>Critical behavior</term>
<term>Differential equations</term>
<term>High temperature</term>
<term>Ising model</term>
<term>Lattice model</term>
<term>Low temperature</term>
<term>Magnetic susceptibility</term>
<term>Power law</term>
<term>Power spectra</term>
<term>Singularity</term>
<term>Spectrum analysis</term>
<term>Square lattices</term>
</keywords>
<keywords scheme="Pascal" xml:lang="fr"><term>Susceptibilité magnétique</term>
<term>Réseau carré</term>
<term>Modèle réticulaire</term>
<term>Modèle Ising</term>
<term>Basse température</term>
<term>Haute température</term>
<term>Algorithme</term>
<term>Equation différentielle</term>
<term>Singularité</term>
<term>Loi puissance</term>
<term>Comportement critique</term>
<term>Spectre puissance</term>
<term>Analyse spectre</term>
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<front><div type="abstract" xml:lang="en">We calculate very long low- and high-temperature series for the susceptibility X of the square lattice Ising model as well as very long series for the five-particle contribution X<sup>(5)</sup>
and six-particle contribution X<sup>(6)</sup>
. These calculations have been made possible by the use of highly optimized polynomial time modular algorithms and a total of more than 150 000 CPU hours on computer clusters. The series for / (low- and high-temperature regimes), X<sup>(5)</sup>
and X<sup>(6)</sup>
are now extended to 2000 terms. In addition, for X<sup>(5)</sup>
, 10000 terms of the series are calculated modulo a single prime, and have been used to find the linear ODE satisfied by X<sup>(5)</sup>
modulo a prime. A diff-Padé analysis of the 2000 terms series for X<sup>(5)</sup>
and X<sup>(6)</sup>
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. The exponents at all the singularities of the Fuchsian linear ODE of X<sup>(5)</sup>
and the (as yet unknown) ODE of X<sup>(6)</sup>
are given: they are all rational numbers. We find the presence of singularities at w = 1/2 for the linear ODE of X<sup>(5)</sup>
, and w<sup>2</sup>
= 1/8 for the ODE of X<sup>(6)</sup>
, which are not singularities of the 'physical' X<sup>(5)</sup>
and X<sup>(6)</sup>
, that is to say the series solutions of the ODE's which are analytic at w = 0. Furthermore, analysis of the long series for X<sup>(5)</sup>
(and X<sup>(6)</sup>
) combined with the corresponding long series for the full susceptibility X yields previously conjectured singularities in some X<sup>(n)</sup>
, n ≥ 7. The exponents at all these singularities are also seen to be rational numbers. We also present a mechanism of resummation of the logarithmic singularities of the X<sup>(n)</sup>
leading to the known power-law critical behaviour occurring in the full X, and perform a power spectrum analysis giving strong arguments in favour of the existence of a natural boundary for the full susceptibility X.</div>
</front>
</TEI>
<affiliations><list><country><li>Algérie</li>
<li>Australie</li>
<li>Canada</li>
<li>France</li>
</country>
<region><li>Victoria (État)</li>
<li>Wilaya d'Alger</li>
<li>Île-de-France</li>
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<settlement><li>Alger</li>
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<orgName><li>Université de Melbourne</li>
</orgName>
</list>
<tree><country name="Algérie"><noRegion><name sortKey="Boukraa, S" sort="Boukraa, S" uniqKey="Boukraa S" first="S." last="Boukraa">S. Boukraa</name>
</noRegion>
<name sortKey="Hassani, S" sort="Hassani, S" uniqKey="Hassani S" first="S." last="Hassani">S. Hassani</name>
<name sortKey="Zenine, N" sort="Zenine, N" uniqKey="Zenine N" first="N." last="Zenine">N. Zenine</name>
</country>
<country name="Australie"><region name="Victoria (État)"><name sortKey="Guttmann, A J" sort="Guttmann, A J" uniqKey="Guttmann A" first="A. J." last="Guttmann">A. J. Guttmann</name>
</region>
<name sortKey="Jensen, I" sort="Jensen, I" uniqKey="Jensen I" first="I." last="Jensen">I. Jensen</name>
</country>
<country name="France"><region name="Île-de-France"><name sortKey="Maillard, J M" sort="Maillard, J M" uniqKey="Maillard J" first="J-M." last="Maillard">J-M. Maillard</name>
</region>
</country>
<country name="Canada"><noRegion><name sortKey="Nickel, B" sort="Nickel, B" uniqKey="Nickel B" first="B." last="Nickel">B. Nickel</name>
</noRegion>
</country>
</tree>
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</record>
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