Experimental mathematics on the magnetic susceptibility of the square lattice Ising model
Identifieur interne : 008825 ( Main/Curation ); précédent : 008824; suivant : 008826Experimental 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, France] ; N. Zenine [Algérie]Source :
- Journal of Physics A: Mathematical and Theoretical [ 1751-8113 ] ; 2008.
Descripteurs français
- Pascal (Inist)
English descriptors
- KwdEn :
- Additional singularities, Algorithm, Algorithms, Analogue, Analytic continuation, Ansatz, Apparent singularities, Boukraa, Circle singularities, Constraint, Correct digits, Critical behavior, Dapp, Denominator, Denominator factors, Differential equations, Differential operator, Digit, Digit accuracy, Divisor, Exact values, Exponent, Fermionic, Fermionic factor, Fuchsian, Full susceptibility, Hassani, Head polynomial, High temperature, Indicial, Indicial equation, Indicial equations, Indicial exponents, Integer, Integral singularity, Integrand, Irreducible, Ising, Ising model, Landau, Landau analysis, Landau conditions, Landau singularities, Landau singularity analysis, Lattice model, Local exponent, Local exponents, Logarithmic, Logarithmic singularities, Long series, Low temperature, Magnetic susceptibility, Maillard, Math, Minimum order, Modulo, More terms, Natural boundary, Nickelian, Nickelian singularities, Orrick, Other exponents, Other hand, Other singularities, Partial sums, Phase constraint, Phys, Pinch singularity, Power law, Power spectra, Prime analysis, Principal disc, Rational numbers, Recursion, Series generation, Series multiplication, Sin2, Singularity, Singularity conditions, Singularity exponent, Singularity exponents, Singularity polynomial, Singularity polynomials, Spectrum analysis, Square lattices, Stationary, Stationary condition, Susceptibility, Taylor expansion, Theor, True singularities, Unit circle, Zenine.
- Teeft :
- Additional singularities, Algorithm, Analogue, Analytic continuation, Ansatz, Apparent singularities, Boukraa, Circle singularities, Constraint, Correct digits, Dapp, Denominator, Denominator factors, Differential operator, Digit, Digit accuracy, Divisor, Exact values, Exponent, Fermionic, Fermionic factor, Fuchsian, Full susceptibility, Hassani, Head polynomial, Indicial, Indicial equation, Indicial equations, Indicial exponents, Integer, Integral singularity, Integrand, Irreducible, Ising, Ising model, Landau, Landau analysis, Landau conditions, Landau singularities, Landau singularity analysis, Local exponent, Local exponents, Logarithmic, Logarithmic singularities, Long series, Magnetic susceptibility, Maillard, Math, Minimum order, Modulo, More terms, Natural boundary, Nickelian, Nickelian singularities, Orrick, Other exponents, Other hand, Other singularities, Partial sums, Phase constraint, Phys, Pinch singularity, Prime analysis, Principal disc, Rational numbers, Recursion, Series generation, Series multiplication, Sin2, Singularity, Singularity conditions, Singularity exponent, Singularity exponents, Singularity polynomial, Singularity polynomials, Stationary, Stationary condition, Susceptibility, Taylor expansion, Theor, True singularities, Unit circle, Zenine.
Abstract
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 (5) and six-particle contribution (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), (5) and (6) are now extended to 2000 terms. In addition, for (5), 10000 terms of the series are calculated modulo a single prime, and have been used to find the linear ODE satisfied by (5) modulo a prime. A diff-Pad analysis of the 2000 terms series for (5) and (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 (5) and the (as yet unknown) ODE of (6) are given: they are all rational numbers. We find the presence of singularities at w 1/2 for the linear ODE of (5), and w2 1/8 for the ODE of (6), which are not singularities of the physical (5) and (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 (5) (and (6)) combined with the corresponding long series for the full susceptibility yields previously conjectured singularities in some (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 (n) leading to the known power-law critical behaviour occurring in the full , and perform a power spectrum analysis giving strong arguments in favour of the existence of a natural boundary for the full susceptibility .
Url:
DOI: 10.1088/1751-8113/41/45/455202
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<front><div type="abstract">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 (5) and six-particle contribution (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), (5) and (6) are now extended to 2000 terms. In addition, for (5), 10000 terms of the series are calculated modulo a single prime, and have been used to find the linear ODE satisfied by (5) modulo a prime. A diff-Pad analysis of the 2000 terms series for (5) and (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 (5) and the (as yet unknown) ODE of (6) are given: they are all rational numbers. We find the presence of singularities at w 1/2 for the linear ODE of (5), and w2 1/8 for the ODE of (6), which are not singularities of the physical (5) and (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 (5) (and (6)) combined with the corresponding long series for the full susceptibility yields previously conjectured singularities in some (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 (n) leading to the known power-law critical behaviour occurring in the full , and perform a power spectrum analysis giving strong arguments in favour of the existence of a natural boundary for the full susceptibility .</div>
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</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>
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</author>
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<s2>Guelph, Ontario NIG 2W1</s2>
<s3>CAN</s3>
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<author><name sortKey="Zenine, N" sort="Zenine, N" uniqKey="Zenine N" first="N." last="Zenine">N. Zenine</name>
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<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>
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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>
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</affiliation>
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<wicri:noRegion>Ontario N1G 2W1</wicri:noRegion>
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<affiliation wicri:level="1"><country xml:lang="fr">Algérie</country>
<wicri:regionArea>Centre de Recherche Nuclaire d'Alger, 2 Bd. Frantz Fanon, BP 399, 16000 Alger</wicri:regionArea>
<placeName><settlement type="city">Alger</settlement>
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<profileDesc><textClass><keywords scheme="KwdEn" xml:lang="en"><term>Additional singularities</term>
<term>Algorithm</term>
<term>Analogue</term>
<term>Analytic continuation</term>
<term>Ansatz</term>
<term>Apparent singularities</term>
<term>Boukraa</term>
<term>Circle singularities</term>
<term>Constraint</term>
<term>Correct digits</term>
<term>Dapp</term>
<term>Denominator</term>
<term>Denominator factors</term>
<term>Differential operator</term>
<term>Digit</term>
<term>Digit accuracy</term>
<term>Divisor</term>
<term>Exact values</term>
<term>Exponent</term>
<term>Fermionic</term>
<term>Fermionic factor</term>
<term>Fuchsian</term>
<term>Full susceptibility</term>
<term>Hassani</term>
<term>Head polynomial</term>
<term>Indicial</term>
<term>Indicial equation</term>
<term>Indicial equations</term>
<term>Indicial exponents</term>
<term>Integer</term>
<term>Integral singularity</term>
<term>Integrand</term>
<term>Irreducible</term>
<term>Ising</term>
<term>Ising model</term>
<term>Landau</term>
<term>Landau analysis</term>
<term>Landau conditions</term>
<term>Landau singularities</term>
<term>Landau singularity analysis</term>
<term>Local exponent</term>
<term>Local exponents</term>
<term>Logarithmic</term>
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<term>Long series</term>
<term>Magnetic susceptibility</term>
<term>Maillard</term>
<term>Math</term>
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<term>More terms</term>
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<term>Phase constraint</term>
<term>Phys</term>
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<term>Prime analysis</term>
<term>Principal disc</term>
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<term>Recursion</term>
<term>Series generation</term>
<term>Series multiplication</term>
<term>Sin2</term>
<term>Singularity</term>
<term>Singularity conditions</term>
<term>Singularity exponent</term>
<term>Singularity exponents</term>
<term>Singularity polynomial</term>
<term>Singularity polynomials</term>
<term>Stationary</term>
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<term>Susceptibility</term>
<term>Taylor expansion</term>
<term>Theor</term>
<term>True singularities</term>
<term>Unit circle</term>
<term>Zenine</term>
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<term>Algorithm</term>
<term>Analogue</term>
<term>Analytic continuation</term>
<term>Ansatz</term>
<term>Apparent singularities</term>
<term>Boukraa</term>
<term>Circle singularities</term>
<term>Constraint</term>
<term>Correct digits</term>
<term>Dapp</term>
<term>Denominator</term>
<term>Denominator factors</term>
<term>Differential operator</term>
<term>Digit</term>
<term>Digit accuracy</term>
<term>Divisor</term>
<term>Exact values</term>
<term>Exponent</term>
<term>Fermionic</term>
<term>Fermionic factor</term>
<term>Fuchsian</term>
<term>Full susceptibility</term>
<term>Hassani</term>
<term>Head polynomial</term>
<term>Indicial</term>
<term>Indicial equation</term>
<term>Indicial equations</term>
<term>Indicial exponents</term>
<term>Integer</term>
<term>Integral singularity</term>
<term>Integrand</term>
<term>Irreducible</term>
<term>Ising</term>
<term>Ising model</term>
<term>Landau</term>
<term>Landau analysis</term>
<term>Landau conditions</term>
<term>Landau singularities</term>
<term>Landau singularity analysis</term>
<term>Local exponent</term>
<term>Local exponents</term>
<term>Logarithmic</term>
<term>Logarithmic singularities</term>
<term>Long series</term>
<term>Magnetic susceptibility</term>
<term>Maillard</term>
<term>Math</term>
<term>Minimum order</term>
<term>Modulo</term>
<term>More terms</term>
<term>Natural boundary</term>
<term>Nickelian</term>
<term>Nickelian singularities</term>
<term>Orrick</term>
<term>Other exponents</term>
<term>Other hand</term>
<term>Other singularities</term>
<term>Partial sums</term>
<term>Phase constraint</term>
<term>Phys</term>
<term>Pinch singularity</term>
<term>Prime analysis</term>
<term>Principal disc</term>
<term>Rational numbers</term>
<term>Recursion</term>
<term>Series generation</term>
<term>Series multiplication</term>
<term>Sin2</term>
<term>Singularity</term>
<term>Singularity conditions</term>
<term>Singularity exponent</term>
<term>Singularity exponents</term>
<term>Singularity polynomial</term>
<term>Singularity polynomials</term>
<term>Stationary</term>
<term>Stationary condition</term>
<term>Susceptibility</term>
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<front><div type="abstract">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 (5) and six-particle contribution (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), (5) and (6) are now extended to 2000 terms. In addition, for (5), 10000 terms of the series are calculated modulo a single prime, and have been used to find the linear ODE satisfied by (5) modulo a prime. A diff-Pad analysis of the 2000 terms series for (5) and (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 (5) and the (as yet unknown) ODE of (6) are given: they are all rational numbers. We find the presence of singularities at w 1/2 for the linear ODE of (5), and w2 1/8 for the ODE of (6), which are not singularities of the physical (5) and (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 (5) (and (6)) combined with the corresponding long series for the full susceptibility yields previously conjectured singularities in some (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 (n) leading to the known power-law critical behaviour occurring in the full , and perform a power spectrum analysis giving strong arguments in favour of the existence of a natural boundary for the full susceptibility .</div>
</front>
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