Experimental mathematics on the magnetic susceptibility of the square lattice Ising model
Identifieur interne : 002E60 ( PascalFrancis/Curation ); précédent : 002E59; suivant : 002E61Experimental 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.
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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<term>Magnetic susceptibility</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>36</s5>
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<s5>37</s5>
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<s5>37</s5>
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<s5>38</s5>
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<s5>38</s5>
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