AustralieFrV1, PascalFrancis, Checkpoint, bibRecord, 003244

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

Identifieur interne : 003244 ( PascalFrancis/Checkpoint ); précédent : 003243; suivant : 003245

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.

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<term>Haute température</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>
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<fC01 i1="01" l="ENG"><s0>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.</s0>
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<s5>27</s5>
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<s5>27</s5>
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<s5>28</s5>
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<s5>28</s5>
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<s5>28</s5>
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<s5>29</s5>
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<s5>30</s5>
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<s5>32</s5>
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<s5>32</s5>
</fC03>
<fC03 i1="08" i2="3" l="FRE"><s0>Equation différentielle</s0>
<s5>33</s5>
</fC03>
<fC03 i1="08" i2="3" l="ENG"><s0>Differential equations</s0>
<s5>33</s5>
</fC03>
<fC03 i1="09" i2="3" l="FRE"><s0>Singularité</s0>
<s5>34</s5>
</fC03>
<fC03 i1="09" i2="3" l="ENG"><s0>Singularity</s0>
<s5>34</s5>
</fC03>
<fC03 i1="10" i2="X" l="FRE"><s0>Loi puissance</s0>
<s5>35</s5>
</fC03>
<fC03 i1="10" i2="X" l="ENG"><s0>Power law</s0>
<s5>35</s5>
</fC03>
<fC03 i1="10" i2="X" l="SPA"><s0>Ley poder</s0>
<s5>35</s5>
</fC03>
<fC03 i1="11" i2="X" l="FRE"><s0>Comportement critique</s0>
<s5>36</s5>
</fC03>
<fC03 i1="11" i2="X" l="ENG"><s0>Critical behavior</s0>
<s5>36</s5>
</fC03>
<fC03 i1="11" i2="X" l="SPA"><s0>Comportamiento crítico</s0>
<s5>36</s5>
</fC03>
<fC03 i1="12" i2="3" l="FRE"><s0>Spectre puissance</s0>
<s5>37</s5>
</fC03>
<fC03 i1="12" i2="3" l="ENG"><s0>Power spectra</s0>
<s5>37</s5>
</fC03>
<fC03 i1="13" i2="X" l="FRE"><s0>Analyse spectre</s0>
<s5>38</s5>
</fC03>
<fC03 i1="13" i2="X" l="ENG"><s0>Spectrum analysis</s0>
<s5>38</s5>
</fC03>
<fC03 i1="13" i2="X" l="SPA"><s0>Análisis espectro</s0>
<s5>38</s5>
</fC03>
<fN21><s1>343</s1>
</fN21>
<fN44 i1="01"><s1>OTO</s1>
</fN44>
<fN82><s1>OTO</s1>
</fN82>
</pA>
</standard>
</inist>
<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>

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

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

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