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

Identifieur interne : 009086 ( Main/Merge ); précédent : 009085; suivant : 009087

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, France] ; N. Zenine [Algérie]

Source :

Journal of Physics A: Mathematical and Theoretical [ 1751-8113 ] ; 2008.

RBID : ISTEX:05A356DDFB8C1D0CFF47BFBBCAA9E229D1E5CD00

English descriptors

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 .

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DOI: 10.1088/1751-8113/41/45/455202

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<profileDesc><textClass><keywords scheme="KwdEn" xml:lang="en"><term>Additional singularities</term>
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<term>Analogue</term>
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<term>Ansatz</term>
<term>Apparent singularities</term>
<term>Boukraa</term>
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<term>Correct digits</term>
<term>Dapp</term>
<term>Denominator</term>
<term>Denominator factors</term>
<term>Differential operator</term>
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<term>Digit accuracy</term>
<term>Divisor</term>
<term>Exact values</term>
<term>Exponent</term>
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<term>Full susceptibility</term>
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<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>
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<term>Logarithmic singularities</term>
<term>Long series</term>
<term>Magnetic susceptibility</term>
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<term>Other hand</term>
<term>Other singularities</term>
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<term>Phase constraint</term>
<term>Phys</term>
<term>Pinch singularity</term>
<term>Prime analysis</term>
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<term>Rational numbers</term>
<term>Recursion</term>
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<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>
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<term>Stationary condition</term>
<term>Susceptibility</term>
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<term>Theor</term>
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<term>Zenine</term>
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<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>
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<term>Full susceptibility</term>
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<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>
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<term>Local exponents</term>
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<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>
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<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>
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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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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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Abstract

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