Iwan Jensen - Ising Series

"Extreme" series related to the square lattice Ising model susceptibility.

First a bit of notation. The `natural' variables are s=sinh(2J/kT) and w=s/(2(1+s²)).
However, the series for the full χ are in the high-temperature variable v=s/2 and the low-temperature variable u=1/(2s)².

High-temperature series for χ in variable v.

Low-temperature series for χ in variable u.

High-temperature series for χ in variable w.

Low-temperature series for χ in variable w.

χ⁽³⁾ in variable w.	χ⁽³⁾ in variable s.	Longer series in w mod the prime 32749.
χ⁽⁴⁾ in variable w.	χ⁽⁴⁾ in variable s.
χ⁽⁵⁾ in variable w.	χ⁽⁵⁾ in variable s.	Longer series in w mod the prime 32749.
χ⁽⁶⁾ in variable w.	χ⁽⁶⁾ in variable s.

Some exact results related to the square lattice Ising model susceptibility.

The first two n-particle contributions to the susceptibility are known exactly.
For the next two exact ODEs are known and for n=5 the ODE is known mod a prime.

χ⁽¹⁾ = -2w/(1-4w)

χ⁽²⁾ = 4w⁴ ₂F ₁[5/2,3/2;3;16w²]

ODE for χ⁽³⁾ in variable w.

First column is the order k of the derivative, the second column is j and the third column is the

j'th coefficient p_k,j in the polynomial P_k(w) multiplying the k'th derivative.

ODE for χ⁽⁴⁾ in variable x=w².

First column is the order k of the derivative, the second column is j and the third column is the

j'th coefficient p_k,j in the polynomial P_k(x) multiplying the k'th derivative.

ODE for χ⁽⁵⁾ in variable w mod the prime 32749.

For χ⁽⁵⁾ we express the solution in terms of a linear ODE with polynomial coefficients but using the

diffential operator (wd/dw). The solution is of order n=56 with the degree of the polynomials equal 129.

The data is organised as a list of lists with the first list being the coefficients of P_n(w),

the second list being the coefficients of P_n-1(w) etc. etc. and the last list is the coefficients of P₀(w)

Low-temperature expansions for the square lattice spin-S Ising model.

The various series are available below. For each value of S (1, 3/2, 2, 5/2 or 3) there are series for the spontaneous magnetisation, susceptibility, specific heat and zero-field partition function. The series are given in terms of the variable u=exp[-J/kTS^2]. Note that the susceptibility and specific heat series start at order 4S (all series coefficients at orders below 4S are identically zero).

All series(uuencoded file)
Spin-1

Magnetisation	A010102
Susceptibility	A010115
Specific Heat	A010111
Partition Function	A010107

Spin-3/2

Magnetisation	A010105
Susceptibility	A010118
Specific Heat	A010113
Partition Function	A010110

Spin-2

Magnetisation	A010103
Susceptibility	A010116
Specific Heat	A010112
Partition Function	A010108

Spin-5/2

Magnetisation	A010106
Susceptibility	A010119
Specific Heat	A010114
Partition Function	A010109

Spin-3

Magnetisation	A010104
Susceptibility	A010117
Specific Heat	A030122
Partition Function	A056620

Created:  24 April, 2000
Last modified: 18 September, 2000