A002913 -id:A002913 - 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

A002926 Low temperature series for spin-1/2 Ising magnetic susceptibility on 3-dimensional simple cubic lattice.

Original entry on oeis.org

0, 0, 1, 0, 12, -14, 135, -276, 1520, -4056, 17778, -54392, 213522, -700362, 2601674, -8836812, 31925046, -110323056, 393008712, -1369533048, 4844047090, -16947396000, 59723296431, -209328634116, 736260986208, -2582605180212, 9074182912884

Offset: 1

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. 421.
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).
C. Vohwinkel, personal communication.

Andrei Zabolotskii, Table of n, a(n) for n = 1..32
Gyan Bhanot, Michael Creutz, Ivan Horvath, Jan Lacki and John Weckel, Series expansions without diagrams, Phys. Rev. E, 49 (1994), 2445-2453; arXiv:hep-lat/9303002, 1993.
C. Domb, Ising model, Phase Transitions and Critical Phenomena 3 (1974), 257, 380-381, 384-387, 390-391, 412-423. (Annotated scanned copy)
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.
Steven R. Finch, Lenz-Ising Constants [broken link]
Steven R. Finch, Lenz-Ising Constants [From the Wayback Machine]
A. J. Guttmann and I. G. Enting, Series studies of the Potts model: I. The simple cubic Ising model, J. Phys. A 26 (1993) 807-821.
Elmar Siepmann, Reihenanalysen für das 3-dimensionale Ising-Modell, Westfälische Wilhelms-Universität Münster, 1993. See Eq. (A.13); divide the coefficients by 4 to get this sequence.
M. F. Sykes et al., Derivation of low-temperature expansions for Ising model VI. Three-dimensional lattices - temperature grouping, J. Phys. A 6 (1973), 1507-1516.

Cf. A002913 (high temperature series); other quantities: A002915 (antiferromagnetic susceptibility), A002891 (partition function), A002929 (magnetization); other lattices: A002927 (square), A002924 (f.c.c.), A002925 (b.c.c.).

Corrections and updates from Steven Finch

a(25)-a(27) from Bhanot et al. added by Andrei Zabolotskii, Feb 09 2022

A010556 High temperature series for spin-1/2 Ising magnetic susceptibility on 4D simple cubic lattice.

Original entry on oeis.org

1, 8, 56, 392, 2696, 18536, 126536, 863720, 5873768, 39942184, 271009112, 1838725896, 12457092504, 84392312392, 571140732808, 3865210690888, 26138072412040, 176752645426600, 1194553221342296, 8073068110703880, 54534614510976680

Offset: 0

N. J. A. Sloane

nonn

S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 391-406.

P. Butera and M. Pernici, High-temperature expansions of the higher susceptibilities for the Ising model in general dimension d, Phys. Rev. E 86, 011139 (2012); arXiv:1209.3592 [hep-lat], 2012.
Steven R. Finch, Lenz-Ising Constants [broken link]
Steven R. Finch, Lenz-Ising Constants [From the Wayback Machine]
M. E. Fisher and D. S. Gaunt, Ising model and self-avoiding walks on hypercubical lattices and high density expansions, Phys. Rev. 133 (1964), A224-A239.
D. S. Gaunt, M. F. Sykes and S. McKenzie, Susceptibility and fourth-field derivative of the spin-1/2 Ising model for T > T_c and d = 4, J. Phys. A 12 (1979), 871-877.
M. A. Moore, Critical behavior of the four-dimensional Ising ferromagnet and the breakdown of scaling, Phys. Rev. B 1 (1970), 2238-2240.

Cf. A002906 (2D), A002913 (3D), A010579 (5D), A010580 (6D), A030008 (7D).

Cf. A030046, A010041, A010044, A010047.

a(17) corrected (was 176752645540264), a(18)-a(20) added using Butera & Pernici's formulas by Andrey Zabolotskiy, Aug 08 2022

A002916 High temperature series for spin-1/2 Ising specific heat on 3-dimensional simple cubic lattice.

Original entry on oeis.org

3, 33, 564, 8976, 155124, 2791308, 51382068, 962178084, 18258531348, 350143322088, 6772382631732, 131922552534036, 2585198190891636, 50919899448451512, 1007393565758096820, 20007153991627682124, 398699967207692643924, 7969220499183448073760, 159718349893920279061428

Offset: 0

N. J. A. Sloane

nonn

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).

G. A. Baker, Further application of the Padé approximant method to the Ising and Heisenberg models, Phys. Rev. 129 (1963) 99-102.
Steven R. Finch, Lenz-Ising Constants [broken link]
Steven R. Finch, Lenz-Ising Constants [From the Wayback Machine]
A. J. Guttmann and I. G. Enting, The high-temperature specific heat exponent of the 3-dimensional Ising model, J. Phys. A 27 (1994) 8007-8010.
Index entries for sequences related to specific heat

Equals 3*A001408.

Cf. A002917 (b.c.c.), A002918 (f.c.c.), A001393 (partition function), A010571 (internal energy), A002913 (susceptibility), A002169 (Heisenberg model), A029872 (square, low-temperature).

3 Cases[Import["https://oeis.org/A001408/b001408.txt", "Table"], {, }][[All, 2]] (* Jean-François Alcover, Jan 17 2020 *)

Sum_{n>=0} a(n) * v^(2*n) = (v^2-1) * (-q/2*f(v)^2 - (v^2-1) * f'(v)^2 + f(v) * (2*v*f'(v) + (v^2-1)*f''(v))) / f(v)^2, where f(v) = Sum_{n>=0} A001393(n) * v^(2*n) and q = 6 is the number of nearest neighbors. - Andrey Zabolotskiy, Feb 15 2022

Corrections and updates from Steven Finch

Terms a(13) and beyond from Andrey Zabolotskiy, Feb 15 2022

A010046 High-temperature expansion of Ising model susceptibility chi_4 for cubic lattice.

Original entry on oeis.org

2, 48, 1272, 38784, 1341408, 52186368, 2256454272, 107494477824, 5595152936448, 316081923944448, 19262189406185472, 1259828274265227264, 88026828815690047488, 6544693787367160086528, 515907116666737635459072, 42981965201161894878511104, 3773829205951827807017238528

Offset: 0

N. J. A. Sloane

nonn

P. Butera and M. Pernici, High-temperature expansions of the higher susceptibilities for the Ising model in general dimension d, Phys. Rev. E 86, 011139 (2012); arXiv:1209.3592 [hep-lat], 2012. Appendix A gives chi_4 as a function of K, which is the negated e.g.f. of this sequence at d=3.
M. Lüscher and P. Weisz, Application of the linked cluster expansion to the n-component phi^4 theory, Nuclear Physics B 300 (1988), 325-359.

Cf. A010045 (square), A010047 (4D cubic), A010040 (chi_2, see also A002913), A010043 (mu_2).

Name clarified, a(14)-a(16) using Butera & Pernici's formulas added by Andrey Zabolotskiy, Nov 25 2024

A010579 High temperature series for spin-1/2 Ising magnetic susceptibility on 5D simple cubic lattice.

Original entry on oeis.org

1, 10, 90, 810, 7210, 64170, 568970, 5044810, 44649930, 395180650, 3494051130, 30893156970, 272971707930, 2411975074570, 21302972395370, 188151452434090, 1661273238131050, 14668124524584170, 129481802727508250, 1142991284620073450, 10087904498275867530

Offset: 0

N. J. A. Sloane

nonn

S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 391-406.

P. Butera and M. Pernici, High-temperature expansions of the higher susceptibilities for the Ising model in general dimension d, Phys. Rev. E 86, 011139 (2012).
Steven R. Finch, Lenz-Ising Constants [broken link]
Steven R. Finch, Lenz-Ising Constants [From the Wayback Machine]
M. E. Fisher and D. S. Gaunt, Ising model and self-avoiding walks on hypercubical lattices and high density expansions, Phys. Rev. 133 (1964), A224-A239.
Misha Gofman, Joan Adler, Amnon Aharony, A. B. Harris and Dietrich Stauffer, Series and Monte Carlo study of high-dimensional Ising models, J. Stat. Phys. 71, 1221-1230 (1993).

Cf. A002906 (2D), A002913 (3D), A010556 (4D), A010580 (6D), A030008 (7D).

Corrections and updates from Steven Finch

Terms a(16)-a(20) added using Butera & Pernici's formulas by Andrey Zabolotskiy, Aug 09 2022

A010580 High temperature series for spin-1/2 Ising magnetic susceptibility on 6D simple cubic lattice.

Original entry on oeis.org

1, 12, 132, 1452, 15852, 173052, 1884972, 20532252, 223437852, 2431526492, 26447593812, 287669976492, 3128064123732, 34013987172972, 369792173040492, 4020299656610636, 43702216875039660, 475060467524653980, 5163624600479230260, 56125562454502452780, 610010748386503122684

Offset: 0

N. J. A. Sloane

nonn

S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 391-406.

P. Butera and M. Pernici, High-temperature expansions of the higher susceptibilities for the Ising model in general dimension d, Phys. Rev. E 86, 011139 (2012).
Steven R. Finch, Lenz-Ising Constants [broken link]
Steven R. Finch, Lenz-Ising Constants [From the Wayback Machine]
M. E. Fisher and D. S. Gaunt, Ising model and self-avoiding walks on hypercubical lattices and high density expansions, Phys. Rev. 133 (1964), A224-A239.
Misha Gofman, Joan Adler, Amnon Aharony, A. B. Harris and Dietrich Stauffer, Series and Monte Carlo study of high-dimensional Ising models, J. Stat. Phys. 71, 1221-1230 (1993).

Cf. A002906 (2D), A002913 (3D), A010556 (4D), A010579 (5D), A030008 (7D).

Corrections and updates from Steven Finch

Terms a(16)-a(20) added using Butera & Pernici's formulas by Andrey Zabolotskiy, Aug 09 2022

A002921 Susceptibility series for f.c.c. lattice.

Original entry on oeis.org

1, 12, 132, 1404, 14652, 151116, 1546332, 15734460, 159425580, 1609987708, 16215457188, 162961837500, 1634743178420, 16373484437340, 163778159931180, 1636328839130860

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.
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).

C. Domb, Self-avoiding walks and the Ising and Heisenberg models, Solid State Physics = Journal of Physics C, 3 (1970) 256-284.
C. Domb, Ising model, Phase Transitions and Critical Phenomena 3 (1974), 257, 380-381, 384-387, 390-391, 412-423. (Annotated scanned copy)
M. E. Fisher and R. J. Burford, Theory of critical point scattering and correlations I: the Ising model, Phys. Rev. 156 (1967), 583-621.
D. S. Gaunt, High Temperature Series Analysis for the Three-Dimensional Ising Model: A Review of Some Recent Work, in: Phase Transitions: Cargèse 1980, eds. Maurice Lévy, Jean-Claude Le Guillou and Jean Zinn-Justin, Springer, Boston, MA, 1982.
Sati McKenzie, High-temperature reduced susceptibility of the Ising model, J. Phys. A: Math. Gen., 8 (1975), L102-L105.
N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 5.
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.
Index entries for sequences related to f.c.c. lattice

Cf. A002913 (cubic), A002914 (b.c.c.); A002918 (specific heat).

Cf. Heisenberg model: A002166; A002170 (cubic), A002168 (b.c.c.); A002165 (specific heat).

a(13)-a(15) from McKenzie added by Andrey Zabolotskiy, Feb 05 2022

A010040 High-temperature expansion of Ising model susceptibility chi_2 for cubic lattice.

Original entry on oeis.org

1, 6, 60, 888, 16944, 403296, 11342400, 371718528, 13814409984, 577113570816, 26660675466240, 1354001083680768, 74774085533159424, 4471493494982516736, 287295290091420794880, 19770508871470350532608, 1448690057046948138123264, 112756550398510017376813056

Offset: 0

N. J. A. Sloane

nonn

M. Lüscher and P. Weisz, Application of the linked cluster expansion to the n-component phi^4 theory, Nuclear Physics B 300 (1988), 325-359.

Cf. A002913, A010039 (square), A010041 (4D cubic), A010043 (mu_2), A010046 (chi_4), A002170 (Heisenberg).

E.g.f.: F(tanh(x)), where F(x) is the g.f. of A002913. - Andrey Zabolotskiy, Nov 19 2024

Name clarified, terms a(15) and beyond using data from A002913 added by Andrey Zabolotskiy, Nov 25 2024

A030008 High temperature series for spin-1/2 Ising magnetic susceptibility on 7D simple cubic lattice.

Original entry on oeis.org

1, 14, 182, 2366, 30590, 395486, 5105870, 65919182, 850586702, 10975573182, 141586912166, 1826501185054, 23558885899318, 303871575267918, 3919114007263518, 50545912921275198, 651868436561980638, 8406864950367314046, 108415583649894484278, 1398136240855886669662, 18029995593288775967598

Offset: 0

Steven Finch

nonn

S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 391-406.

P. Butera and M. Pernici, High-temperature expansions of the higher susceptibilities for the Ising model in general dimension d, Phys. Rev. E 86, 011139 (2012).
Steven R. Finch, Lenz-Ising Constants [broken link]
Steven R. Finch, Lenz-Ising Constants [From the Wayback Machine]
M. E. Fisher and D. S. Gaunt, Ising model and self-avoiding walks on hypercubical lattices and high density expansions, Phys. Rev. 133 (1964), A224-A239.
Misha Gofman, Joan Adler, Amnon Aharony, A. B. Harris and Dietrich Stauffer, Series and Monte Carlo study of high-dimensional Ising models, J. Stat. Phys. 71, 1221-1230 (1993).

Cf. A002906 (2D), A002913 (3D), A010556 (4D), A010579 (5D), A010580 (6D).

Terms a(16)-a(20) added using Butera & Pernici's formulas by Andrey Zabolotskiy, Aug 09 2022

cp's OEIS Frontend

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

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A002926 Low temperature series for spin-1/2 Ising magnetic susceptibility on 3-dimensional simple cubic lattice.

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A010556 High temperature series for spin-1/2 Ising magnetic susceptibility on 4D simple cubic lattice.

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A002916 High temperature series for spin-1/2 Ising specific heat on 3-dimensional simple cubic lattice.

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A010046 High-temperature expansion of Ising model susceptibility chi_4 for cubic lattice.

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A010579 High temperature series for spin-1/2 Ising magnetic susceptibility on 5D simple cubic lattice.

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A010580 High temperature series for spin-1/2 Ising magnetic susceptibility on 6D simple cubic lattice.

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A002921 Susceptibility series for f.c.c. lattice.

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A010040 High-temperature expansion of Ising model susceptibility chi_2 for cubic lattice.

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A030008 High temperature series for spin-1/2 Ising magnetic susceptibility on 7D simple cubic lattice.

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