zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Square lattice Ising model susceptibility: series expansion method and differential equation for χ (3) . (English) Zbl 1113.82020

Summary: In a previous paper [J. Phys. A 37, No. 41, 9651–9668 (2004; Zbl 1073.82014)] we gave the Fuchsian linear differential equation satisfied by χ (3) , the ‘three-particle’ contribution to the susceptibility of the isotropic square lattice Ising model. This paper gives the details of the calculations (with some useful tricks and tools) which allow one to obtain a long series in polynomial time. The method is based on series expansion in the variables that appear in the (n-1)-dimensional integrals representing the n-particle contribution to the isotropic square lattice Ising model susceptibility χ. The integration rules are straightforward due to remarkable formulae we derive for these variables.

We obtain without any numerical approximation χ (3) as a fully integrated series in the variable w=s/2/(1+s 2 ), where s=sinh(2K), with K=J/kT the conventional Ising model coupling constant. We also give some perspectives and comments on these results.

82B27Critical phenomena (equilibrium statistical mechanics)
34M55Painlevé and other special equations; classification, hierarchies
82B10Quantum equilibrium statistical mechanics (general)