A002920 -id:A002920 - cp's OEIS frontend

A002906 High temperature series for spin-1/2 Ising magnetic susceptibility on 2D square lattice.

1, 4, 12, 36, 100, 276, 740, 1972, 5172, 13492, 34876, 89764, 229628, 585508, 1486308, 3763460, 9497380, 23918708, 60080156, 150660388, 377009364, 942106116, 2350157268, 5855734740, 14569318492, 36212402548, 89896870204

Offset: 0

N. J. A. Sloane

The zero-field susceptibility per spin is m^2/kT * Sum_{n >= 0} a(n) * v^n, where v = tanh(J/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.) The b-file has been obtained from the series by Guttmann and Jensen via the substitution t = v/(1-v^2). - Andrey Zabolotskiy, Feb 11 2022

C. Domb, Ising model, in Phase Transitions and Critical Phenomena, vol. 3, ed. C. Domb and M. S. Green, Academic Press, 1974; p. 380.
S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 391-406.
A. J. Guttmann, Asymptotic analysis of power-series expansions, pp. 1-234 of C. Domb and J. L. Lebowitz, editors, Phase Transitions and Critical Phenomena. Vol. 13, Academic Press, NY, 1989.
B. G. Nickel, personal communication.
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).

Andrey Zabolotskiy, Table of n, a(n) for n = 0..2043 (terms up to n = 116 from Fred Hucht)
C. Domb, Ising model, Phase Transitions and Critical Phenomena 3 (1974), 257, 380-381, 384-387, 390-391, 412-423. (Annotated scanned copy)
Steven R. Finch, Lenz-Ising Constants [broken link]
Steven R. Finch, Lenz-Ising Constants [From the Wayback Machine]
M. E. Fisher and R. J. Burford, Theory of critical point scattering and correlations I: the Ising model, Phys. Rev. 156 (1967), 583-621.
S. Gartenhaus and W. S. McCullough, Higher order corrections for the quadratic Ising lattice susceptibility at criticality, Phys. Rev. B 38 (1988) 11688-11703.
A. J. Guttmann, Asymptotic analysis of power-series expansions, pp. 1-13, 56-57, 142-143, 150-151 from of C. Domb and J. L. Lebowitz, editors, Phase Transitions and Critical Phenomena. Vol. 13, Academic Press, NY, 1989. (Annotated scanned copy)
Tony Guttmann, Homepage. See Numerical Data, Ising square lattice susceptibility series, High temperature series.
Iwan Jensen, Series for the Ising model
B. Nickel, On the singularity structure of the 2D Ising model susceptibility, Journal of Physics A, Math. Gen. 32, 3889 (1999); Addendum, 33, 1693 (2000).
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.
M. F. Sykes et al., The asymptotic behavior of selfavoiding walks and returns on a lattice, J. Phys. A 5 (1972), 653-660.
Peter Young, Coefficients in the series expansions

Cf. A002927 (low-temperature), A002908 (energy), A002920 (hexagonal lattice), A002910 (honeycomb), A002913 (cubic lattice), A005401 (Heisenberg).

a(n) ~ c * n^(3/4) * (1 + sqrt(2))^n, where c = 0.839697019... - Vaclav Kotesovec, May 04 2024

Corrections and updates from Steven Finch

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Mar 01 2008

A002913 High temperature series for spin-1/2 Ising magnetic susceptibility on 3-dimensional simple cubic lattice.

Original entry on oeis.org

1, 6, 30, 150, 726, 3510, 16710, 79494, 375174, 1769686, 8306862, 38975286, 182265822, 852063558, 3973784886, 18527532310, 86228667894, 401225368086, 1864308847838, 8660961643254, 40190947325670, 186475398518726, 864404776466406, 4006394107568934, 18554916271112254, 85923704942057238

Offset: 0

N. J. A. Sloane

C. Domb, Ising model, in Phase Transitions and Critical Phenomena, vol. 3, ed. C. Domb and M. S. Green, Academic Press, 1974; p. 381.
S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 391-406.
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).

Andrey Zabolotskiy, Table of n, a(n) for n = 0..32 (terms a(24), a(25) taken from the Campostrini et al. 2002 article by _Per H. Lundow_, terms a(26)-a(32) taken from the Toshiaki Fujiwara and Hiroaki Arisue's slides)
P. Butera and M. Comi, N-vector spin models on the simple-cubic and the body-centered-cubic lattices: A study of the critical behavior of the susceptibility and of the correlation length by high-temperature series extended to order beta^21, Phys. Rev. B 56 (1997) 8212-8240; arXiv:hep-lat/9703018, 1997.
P. Butera and M. Comi, Extension to order b23 of the high-temperature expansions for the spin-1/2 Ising model on the simple-cubic and the body-centered-cubic lattices, BICOCCA/FT-00-09 (June 2000). Phys. Rev. B62 (2000) 14837-14843.
M. Campostrini, Linked-Cluster Expansion of the Ising Model, Journal of Statistical Physics, 103 (2001), 369-394.
M. Campostrini, A. Pelissetto, P. Rossi, and E. Vicari, 25th-order high-temperature expansion results for three-dimensional Ising-like systems on the simple-cubic lattice, Phys. Rev. E, 65 (2002), 66-127.
C. Domb, Ising model, Phase Transitions and Critical Phenomena 3 (1974), 257, 380-381, 384-387, 390-391, 412-423. (Annotated scanned copy)
Steven R. Finch, Lenz-Ising Constants [broken link]
Steven R. Finch, Lenz-Ising Constants [From the Wayback Machine]
M. E. Fisher and R. J. Burford, Theory of critical point scattering and correlations I: the Ising model, Phys. Rev. 156 (1967), 583-621.
Toshiaki Fujiwara and Hiroaki Arisue (presenter), ３次元イジング模型の高温展開 (High-temperature expansion for the 3D Ising model), Computational Physics with CP-PACS 2002 Workshop [in Japanese].
Toshiaki Fujiwara and Hiroaki Arisue (presenter), New algorithm of the high-temperature expansion for the Ising model in three dimensions, Asia-Pacific Mini-Workshop on Lattice QCD, Center for Computational Physics, University of Tsukuba, 2003: abstract, slides, source.
D. S. Gaunt, High Temperature Series Analysis for the Three-Dimensional Ising Model: A Review of Some Recent Work, pp. 217-246 in: Phase Transitions: Cargèse 1980, eds. Maurice Lévy, Jean-Claude Le Guillou and Jean Zinn-Justin, Springer, Boston, MA, 1982.
M. F. Sykes, D. G. Gaunt, P. D. Roberts and J. A. Wyles, High temperature series for the susceptibility of the Ising model, II. Three dimensional lattices, J. Phys. A 5 (1972) 640-652.

Cf. other quantities: A001393 (partition function), A010571 (internal energy), A002916 (specific heat), A003490 (surface susceptibility), A007287 (layer susceptibility), A010040, A010043, A010046.
Cf. other structures: A002906 (square), A002920 (hexagonal), A002910 (honeycomb), A002914 (b.c.c.), A002921 (f.c.c.), A003119 (diamond), A010556 (4D cubic), A010579 (5D cubic), A010580 (6D cubic), A030008 (7D cubic).
Cf. low-temperature series: A002926 (ferromagnetic), A002915 (antiferromagnetic).
Cf. other models: A002170 (Heisenberg), A003279 (spherical).

Corrections and updates from Steven Finch
More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Mar 01 2008
Several errors in the sequence were corrected by Per H. Lundow, Jan 17 2011

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

Original entry on oeis.org

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

A005399 E.g.f.: high-temperature series in J/2kT for ferromagnetic susceptibility for the spin-1/2 Heisenberg model on hexagonal lattice.

Original entry on oeis.org

1, 6, 48, 408, 3600, 42336, 781728, 13646016, 90893568, -1798204416, 70794720768, 7538546211840, 63813109782528, -12977417912045568, -320549902414196736, 33016479733605777408, 1709506241695601983488

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.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. A. Baker Jr., H. E. Gilbert, J. Eve, and G. S. Rushbrooke, On the two-dimensional, spin-1/2 Heisenberg ferromagnetic models, Phys. Lett., 25A (1967), 207-209.
N. Elstner, R. R. P. Singh and A. P. Young, Finite temperature properties of the spin-1/2 Heisenberg antiferromagnet on the triangular lattice, Phys. Rev. Lett., 71 (1993), 1629-1632.
G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2
J. Oitmaa and E. Bornilla, High-temperature-series study of the spin-1/2 Heisenberg ferromagnet, Phys. Rev. B, 53 (1996), 14228.
Laurent Pierre, Bernard Bernu and Laura Messio, High temperature series expansions of S = 1/2 Heisenberg spin models: Algorithm to include the magnetic field with optimized complexity, SciPost Phys. 17, 105 (2024); arXiv:2404.02271 [cond-mat.str-el], 2024. See the supporting file Triangle_16_16.py; multiply pol1[1] by 2 to get this sequence.

Cf. A002920 (Ising high-temperature), A047709 (Ising low-temperature), A005400 (series for specific heat, or free energy), A005401 (square lattice), A005402 (specific heat for square lattice).

New name from Andrey Zabolotskiy, Mar 03 2021
a(10)-a(12) added from Oitmaa and Bornilla by Andrey Zabolotskiy, Oct 20 2021
a(0) and a(13) using data from Elstner et al. (see Table I for the values -(-1)^n*n*a(n-1)) added by Andrey Zabolotskiy, Jun 17 2022
a(14)-a(16) using Pierre, Bernu & Messio's data added by Andrey Zabolotskiy, Nov 25 2024

A002919 High-temperature series for susceptibility for the spin-1/2 Ising model on hexagonal lattice.

Original entry on oeis.org

1, 6, 24, 90, 318, 1098, 3696, 12270, 40224, 130650, 421176, 1348998, 4299018, 13635630, 43092888, 135698970, 426144654, 1334488074, 4170038328, 13001153910, 40464412482, 125706293478, 389962873920, 1207855307874, 3736709089176, 11544946664622, 35633199126576

Offset: 0

N. J. A. Sloane

Previous name was: Susceptibility 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.

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).

G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2
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, A002920, A047709.

a(n) = A002910(2*n), cf. A002920. - Andrey Zabolotskiy, Mar 01 2021

New name and more terms using A002920 from Andrey Zabolotskiy, Mar 03 2021

cp's OEIS Frontend

A002906 High temperature series for spin-1/2 Ising magnetic susceptibility on 2D square lattice.

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A002913 High temperature series for spin-1/2 Ising magnetic susceptibility on 3-dimensional simple cubic lattice.

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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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A005399 E.g.f.: high-temperature series in J/2kT for ferromagnetic susceptibility for the spin-1/2 Heisenberg model on hexagonal lattice.

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A002919 High-temperature series for susceptibility for the spin-1/2 Ising model on hexagonal lattice.

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