AustralieFrV1, Main, Merge, bibRecord, 009658

Experimental mathematics on the magnetic susceptibility of the square lattice Ising model

Identifieur interne : 009658 ( Main/Merge ); précédent : 009657; suivant : 009659

Experimental 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.

RBID : Pascal:08-0524292

Descripteurs français

Pascal (Inist)
- Susceptibilité magnétique, Réseau carré, Modèle réticulaire, Modèle Ising, Basse température, Haute température, Algorithme, Equation différentielle, Singularité, Loi puissance, Comportement critique, Spectre puissance, Analyse spectre.

English descriptors

KwdEn :
- Algorithms, Critical behavior, Differential equations, High temperature, Ising model, Lattice model, Low temperature, Magnetic susceptibility, Power law, Power spectra, Singularity, Spectrum analysis, Square lattices.

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.

Links toward previous steps (curation, corpus...)

to stream PascalFrancis, to step Corpus: 003193
to stream PascalFrancis, to step Curation: 002E60
to stream PascalFrancis, to step Checkpoint: 003244

Links to Exploration step

Pascal:08-0524292

Le document en format XML

<record><TEI><teiHeader><fileDesc><titleStmt><title xml:lang="en" level="a">Experimental mathematics on the magnetic susceptibility of the square lattice Ising model</title>
<author><name sortKey="Boukraa, S" sort="Boukraa, S" uniqKey="Boukraa S" first="S." last="Boukraa">S. Boukraa</name>
<affiliation wicri:level="1"><inist:fA14 i1="01"><s1>LPTHIRM and Département d'Aéronautique, Université de Blida</s1>
<s3>DZA</s3>
<sZ>1 aut.</sZ>
</inist:fA14>
<country>Algérie</country>
<wicri:noRegion>LPTHIRM and Département d'Aéronautique, Université de Blida</wicri:noRegion>
</affiliation>
</author>
<author><name sortKey="Guttmann, A J" sort="Guttmann, A J" uniqKey="Guttmann A" first="A. J." last="Guttmann">A. J. Guttmann</name>
<affiliation wicri:level="4"><inist:fA14 i1="02"><s1>ARC Centre of Excellence for Mathematics and Statistics of Complex Systems, Department of Mathematics and Statistics, The University of Melbourne</s1>
<s2>Victoria 3010</s2>
<s3>AUS</s3>
<sZ>2 aut.</sZ>
<sZ>4 aut.</sZ>
</inist:fA14>
<country>Australie</country>
<placeName><settlement type="city">Melbourne</settlement>
<region type="état">Victoria (État)</region>
</placeName>
<orgName type="university">Université de Melbourne</orgName>
</affiliation>
</author>
<author><name sortKey="Hassani, S" sort="Hassani, S" uniqKey="Hassani S" first="S." last="Hassani">S. Hassani</name>
<affiliation wicri:level="1"><inist:fA14 i1="03"><s1>Centre de Recherche Nucléaire d'Alger, 2 Bd. Frantz Fanon, BP 399</s1>
<s2>16000 Alger</s2>
<s3>DZA</s3>
<sZ>3 aut.</sZ>
<sZ>7 aut.</sZ>
</inist:fA14>
<country>Algérie</country>
<placeName><settlement type="city">Alger</settlement>
<region nuts="2">Wilaya d'Alger</region>
</placeName>
</affiliation>
</author>
<author><name sortKey="Jensen, I" sort="Jensen, I" uniqKey="Jensen I" first="I." last="Jensen">I. Jensen</name>
<affiliation wicri:level="4"><inist:fA14 i1="02"><s1>ARC Centre of Excellence for Mathematics and Statistics of Complex Systems, Department of Mathematics and Statistics, The University of Melbourne</s1>
<s2>Victoria 3010</s2>
<s3>AUS</s3>
<sZ>2 aut.</sZ>
<sZ>4 aut.</sZ>
</inist:fA14>
<country>Australie</country>
<placeName><settlement type="city">Melbourne</settlement>
<region type="état">Victoria (État)</region>
</placeName>
<orgName type="university">Université de Melbourne</orgName>
</affiliation>
</author>
<author><name sortKey="Maillard, J M" sort="Maillard, J M" uniqKey="Maillard J" first="J-M." last="Maillard">J-M. Maillard</name>
<affiliation wicri:level="3"><inist:fA14 i1="04"><s1>LPTMC, Université de Paris, Tour 24, 4ème étage, case 121, 4 Place Jussieu</s1>
<s2>75252 Paris</s2>
<s3>FRA</s3>
<sZ>5 aut.</sZ>
</inist:fA14>
<country>France</country>
<placeName><region type="region" nuts="2">Île-de-France</region>
<settlement type="city">Paris</settlement>
</placeName>
</affiliation>
</author>
<author><name sortKey="Nickel, B" sort="Nickel, B" uniqKey="Nickel B" first="B." last="Nickel">B. Nickel</name>
<affiliation wicri:level="1"><inist:fA14 i1="05"><s1>Department of Physics, University of Guelph</s1>
<s2>Guelph, Ontario NIG 2W1</s2>
<s3>CAN</s3>
<sZ>6 aut.</sZ>
</inist:fA14>
<country>Canada</country>
<wicri:noRegion>Guelph, Ontario NIG 2W1</wicri:noRegion>
</affiliation>
</author>
<author><name sortKey="Zenine, N" sort="Zenine, N" uniqKey="Zenine N" first="N." last="Zenine">N. Zenine</name>
<affiliation wicri:level="1"><inist:fA14 i1="03"><s1>Centre de Recherche Nucléaire d'Alger, 2 Bd. Frantz Fanon, BP 399</s1>
<s2>16000 Alger</s2>
<s3>DZA</s3>
<sZ>3 aut.</sZ>
<sZ>7 aut.</sZ>
</inist:fA14>
<country>Algérie</country>
<placeName><settlement type="city">Alger</settlement>
<region nuts="2">Wilaya d'Alger</region>
</placeName>
</affiliation>
</author>
</titleStmt>
<publicationStmt><idno type="wicri:source">INIST</idno>
<idno type="inist">08-0524292</idno>
<date when="2008">2008</date>
<idno type="stanalyst">PASCAL 08-0524292 INIST</idno>
<idno type="RBID">Pascal:08-0524292</idno>
<idno type="wicri:Area/PascalFrancis/Corpus">003193</idno>
<idno type="wicri:Area/PascalFrancis/Curation">002E60</idno>
<idno type="wicri:Area/PascalFrancis/Checkpoint">003244</idno>
<idno type="wicri:explorRef" wicri:stream="PascalFrancis" wicri:step="Checkpoint">003244</idno>
<idno type="wicri:doubleKey">1751-8113:2008:Boukraa S:experimental:mathematics:on</idno>
<idno type="wicri:Area/Main/Merge">009658</idno>
</publicationStmt>
<sourceDesc><biblStruct><analytic><title xml:lang="en" level="a">Experimental mathematics on the magnetic susceptibility of the square lattice Ising model</title>
<author><name sortKey="Boukraa, S" sort="Boukraa, S" uniqKey="Boukraa S" first="S." last="Boukraa">S. Boukraa</name>
<affiliation wicri:level="1"><inist:fA14 i1="01"><s1>LPTHIRM and Département d'Aéronautique, Université de Blida</s1>
<s3>DZA</s3>
<sZ>1 aut.</sZ>
</inist:fA14>
<country>Algérie</country>
<wicri:noRegion>LPTHIRM and Département d'Aéronautique, Université de Blida</wicri:noRegion>
</affiliation>
</author>
<author><name sortKey="Guttmann, A J" sort="Guttmann, A J" uniqKey="Guttmann A" first="A. J." last="Guttmann">A. J. Guttmann</name>
<affiliation wicri:level="4"><inist:fA14 i1="02"><s1>ARC Centre of Excellence for Mathematics and Statistics of Complex Systems, Department of Mathematics and Statistics, The University of Melbourne</s1>
<s2>Victoria 3010</s2>
<s3>AUS</s3>
<sZ>2 aut.</sZ>
<sZ>4 aut.</sZ>
</inist:fA14>
<country>Australie</country>
<placeName><settlement type="city">Melbourne</settlement>
<region type="état">Victoria (État)</region>
</placeName>
<orgName type="university">Université de Melbourne</orgName>
</affiliation>
</author>
<author><name sortKey="Hassani, S" sort="Hassani, S" uniqKey="Hassani S" first="S." last="Hassani">S. Hassani</name>
<affiliation wicri:level="1"><inist:fA14 i1="03"><s1>Centre de Recherche Nucléaire d'Alger, 2 Bd. Frantz Fanon, BP 399</s1>
<s2>16000 Alger</s2>
<s3>DZA</s3>
<sZ>3 aut.</sZ>
<sZ>7 aut.</sZ>
</inist:fA14>
<country>Algérie</country>
<placeName><settlement type="city">Alger</settlement>
<region nuts="2">Wilaya d'Alger</region>
</placeName>
</affiliation>
</author>
<author><name sortKey="Jensen, I" sort="Jensen, I" uniqKey="Jensen I" first="I." last="Jensen">I. Jensen</name>
<affiliation wicri:level="4"><inist:fA14 i1="02"><s1>ARC Centre of Excellence for Mathematics and Statistics of Complex Systems, Department of Mathematics and Statistics, The University of Melbourne</s1>
<s2>Victoria 3010</s2>
<s3>AUS</s3>
<sZ>2 aut.</sZ>
<sZ>4 aut.</sZ>
</inist:fA14>
<country>Australie</country>
<placeName><settlement type="city">Melbourne</settlement>
<region type="état">Victoria (État)</region>
</placeName>
<orgName type="university">Université de Melbourne</orgName>
</affiliation>
</author>
<author><name sortKey="Maillard, J M" sort="Maillard, J M" uniqKey="Maillard J" first="J-M." last="Maillard">J-M. Maillard</name>
<affiliation wicri:level="3"><inist:fA14 i1="04"><s1>LPTMC, Université de Paris, Tour 24, 4ème étage, case 121, 4 Place Jussieu</s1>
<s2>75252 Paris</s2>
<s3>FRA</s3>
<sZ>5 aut.</sZ>
</inist:fA14>
<country>France</country>
<placeName><region type="region" nuts="2">Île-de-France</region>
<settlement type="city">Paris</settlement>
</placeName>
</affiliation>
</author>
<author><name sortKey="Nickel, B" sort="Nickel, B" uniqKey="Nickel B" first="B." last="Nickel">B. Nickel</name>
<affiliation wicri:level="1"><inist:fA14 i1="05"><s1>Department of Physics, University of Guelph</s1>
<s2>Guelph, Ontario NIG 2W1</s2>
<s3>CAN</s3>
<sZ>6 aut.</sZ>
</inist:fA14>
<country>Canada</country>
<wicri:noRegion>Guelph, Ontario NIG 2W1</wicri:noRegion>
</affiliation>
</author>
<author><name sortKey="Zenine, N" sort="Zenine, N" uniqKey="Zenine N" first="N." last="Zenine">N. Zenine</name>
<affiliation wicri:level="1"><inist:fA14 i1="03"><s1>Centre de Recherche Nucléaire d'Alger, 2 Bd. Frantz Fanon, BP 399</s1>
<s2>16000 Alger</s2>
<s3>DZA</s3>
<sZ>3 aut.</sZ>
<sZ>7 aut.</sZ>
</inist:fA14>
<country>Algérie</country>
<placeName><settlement type="city">Alger</settlement>
<region nuts="2">Wilaya d'Alger</region>
</placeName>
</affiliation>
</author>
</analytic>
<series><title level="j" type="main">Journal of physics. A, Mathematical and theoretical : (Print)</title>
<title level="j" type="abbreviated">J. phys., A, math. theor. : (Print)</title>
<idno type="ISSN">1751-8113</idno>
<imprint><date when="2008">2008</date>
</imprint>
</series>
</biblStruct>
</sourceDesc>
<seriesStmt><title level="j" type="main">Journal of physics. A, Mathematical and theoretical : (Print)</title>
<title level="j" type="abbreviated">J. phys., A, math. theor. : (Print)</title>
<idno type="ISSN">1751-8113</idno>
</seriesStmt>
</fileDesc>
<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>
</keywords>
</textClass>
</profileDesc>
</teiHeader>
<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>
</region>
<settlement><li>Alger</li>
<li>Melbourne</li>
<li>Paris</li>
</settlement>
<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>
</affiliations>
</record>

Pour manipuler ce document sous Unix (Dilib)

EXPLOR_STEP=$WICRI_ROOT/Wicri/Asie/explor/AustralieFrV1/Data/Main/Merge

HfdSelect -h $EXPLOR_STEP/biblio.hfd -nk 009658 | SxmlIndent | more

HfdSelect -h $EXPLOR_AREA/Data/Main/Merge/biblio.hfd -nk 009658 | SxmlIndent | more

Pour mettre un lien sur cette page dans le réseau Wicri

{{Explor lien
   |wiki=    Wicri/Asie
   |area=    AustralieFrV1
   |flux=    Main
   |étape=   Merge
   |type=    RBID
   |clé=     Pascal:08-0524292
   |texte=   Experimental mathematics on the magnetic susceptibility of the square lattice Ising model
}}

This area was generated with Dilib version V0.6.33.
Data generation: Tue Dec 5 10:43:12 2017. Site generation: Tue Mar 5 14:07:20 2024

	Serveur d'exploration sur les relations entre la France et l'Australie
	Attention, ce site est en cours de développement ! Attention, site généré par des moyens informatiques à partir de corpus bruts. Les informations ne sont donc pas validées.

Serveur d'exploration sur les relations entre la France et l'Australie

Experimental mathematics on the magnetic susceptibility of the square lattice Ising model

Experimental mathematics on the magnetic susceptibility of the square lattice Ising model

Source :

Descripteurs français

English descriptors

Abstract

Links toward previous steps (curation, corpus...)

Links to Exploration step

Le document en format XML

Pour manipuler ce document sous Unix (Dilib)

Pour mettre un lien sur cette page dans le réseau Wicri