A002919 -id:A002919 - cp's OEIS frontend

A047709 Low-temperature series in u = exp(-4J/kT) for ferromagnetic susceptibility for the spin-1/2 Ising model on hexagonal lattice.

0, 0, 1, 0, 12, 4, 129, 72, 1332, 960, 13419, 11372, 132900, 126396, 1299851, 1349784, 12592440, 14023944, 121074183, 142818336, 1157026804, 1432470300, 11001347199, 14196860272, 104161648860, 139351826712, 982653092725, 1357030991292, 9241395939636

Offset: 1

N. J. A. Sloane

The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice.

Many sources give this sequence multiplied by 4 because the actual susceptibility per spin is this series times 4m^2/kT. (m is the magnetic moment of a single spin; the factor m^2 may be present or absent depending on the precise definition of the susceptibility.)

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

Y. Chan, A. J. Guttmann, B. G. Nickel, and J. H. H. Perk, The Ising Susceptibility Scaling Function, J Stat Phys 145 (2011), 549-590; arXiv:1012.5272 [cond-mat.stat-mech], 2010-2020. Gives 642 terms in the file Triangle_u642.txt (divide by 4 to get this sequence).
J. W. Essam and M. E. Fisher, Padé approximant studies of the lattice gas and Ising ferromagnet below the critical point, J. Chem. Phys., 38 (1963), 802-812.
G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2
M. F. Sykes and M. E. Fisher, Antiferromagnetic susceptibility of the plane square and honeycomb Ising lattices, Physica, 28 (1962), 919-938.
M. F. Sykes, D. S. Gaunt, J. L. Martin, S. R. Mattingly, and J. W. Essam, Derivation of low-temperature expansions for Ising model. IV. Two-dimensional lattices--temperature grouping, Journal of Mathematical Physics 14 (1973), 1071.

Cf. A002919, A002920, A003488, A005399, A057383, A057387.

Cf. A002912, A002927, A002924, A002925, A002926, A003220.

Edited and extended from Chan et al by Andrey Zabolotskiy, Mar 02 2021

A002920 High-temperature series in w = tanh(J/kT) for ferromagnetic susceptibility for the spin-1/2 Ising model on hexagonal lattice.

Original entry on oeis.org

1, 6, 30, 138, 606, 2586, 10818, 44574, 181542, 732678, 2935218, 11687202, 46296210, 182588850, 717395262, 2809372302, 10969820358, 42724062966, 166015496838, 643768299018, 2491738141314, 9628130289018, 37146098272266, 143110933254702, 550643544948090

Offset: 0

N. J. A. Sloane

Previous name was: Susceptibility series for hexagonal lattice.
The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice.
The actual susceptibility per spin is this series times m^2/kT. (m is the magnetic moment of a single spin; this factor may be present or absent depending on the precise definition of the susceptibility.)

C. Domb, Ising model, in Phase Transitions and Critical Phenomena, vol. 3, ed. C. Domb and M. S. Green, Academic Press, 1974; p. 380.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Y. Chan, A. J. Guttmann, B. G. Nickel, and J. H. H. Perk, The Ising Susceptibility Scaling Function, J Stat Phys 145 (2011), 549-590; arXiv:1012.5272 [cond-mat.stat-mech], 2010-2020. Gives 320 terms in the file Triangle_v319.
C. Domb, Ising model, Phase Transitions and Critical Phenomena 3 (1974), 257, 380-381, 384-387, 390-391, 412-423. (Annotated scanned copy)
Michael E. Fisher, Transformations of Ising Models, Phys. Rev. 113 (1959), 969-981.
M. E. Fisher and R. J. Burford, Theory of critical point scattering and correlations I: the Ising model, Phys. Rev. 156 (1967), 583-621.
G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2
M. F. Sykes, Some counting theorems in the theory of the Ising problem and the excluded volume problem, J. Math. Phys., 2 (1961), 52-62.
M. F. Sykes, D. G. Gaunt, P. D. Roberts and J. A. Wyles, High temperature series for the susceptibility of the Ising model, I. Two dimensional lattices, J. Phys. A 5 (1972) 624-639.

Cf. A002910, A128834, A047709, A002919.

G.f.: (h(v(w)) + h(-v(w))) / 2, where h(v) is the g.f. of A002910 and v(w)^2 = w*(1+w)/(1+w^3) [Fisher, p. 979]. - Andrey Zabolotskiy, Mar 01 2021

Edited and extended from Chan et al by Andrey Zabolotskiy, Mar 03 2021

cp's OEIS Frontend

A047709 Low-temperature series in u = exp(-4J/kT) for ferromagnetic susceptibility for the spin-1/2 Ising model on hexagonal lattice.

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