Experimental mathematics on the magnetic susceptibility of the square lattice Ising model
Identifieur interne : 003193 ( PascalFrancis/Corpus ); précédent : 003192; suivant : 003194Experimental mathematics on the magnetic susceptibility of the square lattice Ising model
Auteurs : S. Boukraa ; A. J. Guttmann ; S. Hassani ; I. Jensen ; J-M. Maillard ; B. Nickel ; N. ZenineSource :
- Journal of physics. A, Mathematical and theoretical : (Print) [ 1751-8113 ] ; 2008.
Descripteurs français
- 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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NO : | PASCAL 08-0524292 INIST |
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ET : | Experimental mathematics on the magnetic susceptibility of the square lattice Ising model |
AU : | BOUKRAA (S.); GUTTMANN (A. J.); HASSANI (S.); JENSEN (I.); MAILLARD (J-M.); NICKEL (B.); ZENINE (N.) |
AF : | LPTHIRM and Département d'Aéronautique, Université de Blida/Algérie (1 aut.); ARC Centre of Excellence for Mathematics and Statistics of Complex Systems, Department of Mathematics and Statistics, The University of Melbourne/Victoria 3010/Australie (2 aut., 4 aut.); Centre de Recherche Nucléaire d'Alger, 2 Bd. Frantz Fanon, BP 399/16000 Alger/Algérie (3 aut., 7 aut.); LPTMC, Université de Paris, Tour 24, 4ème étage, case 121, 4 Place Jussieu/75252 Paris/France (5 aut.); Department of Physics, University of Guelph/Guelph, Ontario NIG 2W1/Canada (6 aut.) |
DT : | Publication en série; Niveau analytique |
SO : | Journal of physics. A, Mathematical and theoretical : (Print); ISSN 1751-8113; Royaume-Uni; Da. 2008; Vol. 41; No. 45; 455202.1-455202.51; Bibl. 33 ref. |
LA : | Anglais |
EA : | 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. |
CC : | 001B00 |
FD : | Susceptibilité magnétique; Réseau carré; Modèle réticulaire; Modèle Ising; Basse température; Haute température; Algorithme; Equation différentielle; Singularité; Loi puissance; Comportement critique; Spectre puissance; Analyse spectre |
ED : | Magnetic susceptibility; Square lattices; Lattice model; Ising model; Low temperature; High temperature; Algorithms; Differential equations; Singularity; Power law; Critical behavior; Power spectra; Spectrum analysis |
SD : | Modelo reticular; Baja temperatura; Alta temperatura; Ley poder; Comportamiento crítico; Análisis espectro |
LO : | INIST-577C.354000183900970040 |
ID : | 08-0524292 |
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Pascal:08-0524292Le document en format XML
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<term>Lattice model</term>
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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>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>
<server><NO>PASCAL 08-0524292 INIST</NO>
<ET>Experimental mathematics on the magnetic susceptibility of the square lattice Ising model</ET>
<AU>BOUKRAA (S.); GUTTMANN (A. J.); HASSANI (S.); JENSEN (I.); MAILLARD (J-M.); NICKEL (B.); ZENINE (N.)</AU>
<AF>LPTHIRM and Département d'Aéronautique, Université de Blida/Algérie (1 aut.); ARC Centre of Excellence for Mathematics and Statistics of Complex Systems, Department of Mathematics and Statistics, The University of Melbourne/Victoria 3010/Australie (2 aut., 4 aut.); Centre de Recherche Nucléaire d'Alger, 2 Bd. Frantz Fanon, BP 399/16000 Alger/Algérie (3 aut., 7 aut.); LPTMC, Université de Paris, Tour 24, 4ème étage, case 121, 4 Place Jussieu/75252 Paris/France (5 aut.); Department of Physics, University of Guelph/Guelph, Ontario NIG 2W1/Canada (6 aut.)</AF>
<DT>Publication en série; Niveau analytique</DT>
<SO>Journal of physics. A, Mathematical and theoretical : (Print); ISSN 1751-8113; Royaume-Uni; Da. 2008; Vol. 41; No. 45; 455202.1-455202.51; Bibl. 33 ref.</SO>
<LA>Anglais</LA>
<EA>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.</EA>
<CC>001B00</CC>
<FD>Susceptibilité magnétique; Réseau carré; Modèle réticulaire; Modèle Ising; Basse température; Haute température; Algorithme; Equation différentielle; Singularité; Loi puissance; Comportement critique; Spectre puissance; Analyse spectre</FD>
<ED>Magnetic susceptibility; Square lattices; Lattice model; Ising model; Low temperature; High temperature; Algorithms; Differential equations; Singularity; Power law; Critical behavior; Power spectra; Spectrum analysis</ED>
<SD>Modelo reticular; Baja temperatura; Alta temperatura; Ley poder; Comportamiento crítico; Análisis espectro</SD>
<LO>INIST-577C.354000183900970040</LO>
<ID>08-0524292</ID>
</server>
</inist>
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