A002929 -id:A002929 - cp's OEIS frontend

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

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

A002928 Magnetization for square lattice.

Original entry on oeis.org

1, 0, -2, -8, -34, -152, -714, -3472, -17318, -88048, -454378, -2373048, -12515634, -66551016, -356345666, -1919453984, -10392792766, -56527200992, -308691183938, -1691769619240, -9301374102034

Offset: 0

N. J. A. Sloane, Simon Plouffe

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).
J. M. Yeomans, Statistical mechanics of phase transitions, Oxford Univ. Press, 1992, p. 93.

C. Domb, Ising model, Phase Transitions and Critical Phenomena 3 (1974), 257, 380-381, 384-387, 390-391, 412-423. (Annotated scanned copy)
David Park, A summation method for crystal statistics, Physica 22 (1956), 932-940.

Cf. other structures: A007206, A007207, A002929, A002930, A003193, A003196.
Cf. higher spins: A010102, A010105/A030120, A010103, A010106/A030121, A010104.
Cf. Potts model: A057374, A057378.
Cf. A002927 (susceptibility).

series((1+x)^(1/4)*(1-6*x+x^2)^(1/8)/(1-x)^(1/2),x,40).

CoefficientList[Series[(1+x)^(1/4)*(1-6*x+x^2)^(1/8)/(1-x)^(1/2), {x, 0, 20}], x] (* Vaclav Kotesovec, Apr 27 2024 *)

n*a(n) + 6*(-n+1)*a(n-1) + 4*a(n-2) + 6*(n-3)*a(n-3) + (-n+4)*a(n-4) = 0. - R. J. Mathar, Mar 08 2013

a(n) ~ -Gamma(1/8) * (1 + sqrt(2))^(2*n - 1/2) / (Pi * 2^(57/16) * n^(9/8)). - Vaclav Kotesovec, Apr 27 2024

A003193 Magnetization for body-centered cubic lattice.

Original entry on oeis.org

1, 0, 0, 0, -2, 0, 0, -16, 18, 0, -168, 384, -314, -1632, 6264, -9744, -10014, 86976, -205344, 80176, 1009338, -3579568, 4575296, 8301024, -54012882, 112640896, -5164464, -694845120, 2160781086

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. 421.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

C. Domb, Ising model, Phase Transitions and Critical Phenomena 3 (1974), 257, 380-381, 384-387, 390-391, 412-423. (Annotated scanned copy)
M. F. Sykes, J. W. Essam and D. S. Gaunt, Derivation of low-temperature expansions for the Ising model of a ferromagnet and an antiferromagnet, J. Math. Phys. 6 (1965), 283-298.
M. F. Sykes, D. S. Gaunt, J. W. Essam and C. J. Elliott, Derivation of low-temperature expansions for Ising model. VI. Three-dimensional lattices-temperature grouping, J. Phys. A: Math. Nucl. Gen., 6 (1973), 1507-1516.
Index entries for sequences related to b.c.c. lattice

Cf. A002929 (cubic), A003196 (f.c.c.), A002925 (ferromagnetic susceptibility), A003194 (antiferromagnetic susceptibility).

a(27)-a(28) from Sykes et al. (1973) added by Andrey Zabolotskiy, Feb 16 2022

A007270 Low-temperature series for magnetization in zero-field 3-state Potts model on cubic lattice.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, -3, 0, 0, 0, -18, -18, 42, 0, -135, -270, 477, 648, -1980, -2988, 4140, 14052, -21690, -52920, 55020, 201852, -162774, -914538, 555750, 3229524, -1188327, -13301370, 1402686, 52334268, 95751, -195398208, -58983558, 761838084, 359664885, -2910516786, -1946958399, 10681132140, 10207745148, -40522674258

Offset: 0

Simon Plouffe

sign

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..56 (using data from Vohwinkel)
A. J. Guttmann and I. G. Enting, Series studies of the Potts model: III. The 3-state model on the simple cubic lattice, J. Phys. A: Math. Gen., 27 (1994), 5801-5812; arXiv:hep-lat/9312083, 1993. See Table 1; note that row 36 is missing, see Kim.
Seung-Yeon Kim, Partition function zeros of the Q-state Potts model on the simple-cubic lattice, Nuclear Physics B, 637 (2002), 409-426; arXiv:cond-mat/0205451, 2002. See the note to Ref. [58].
S. Miyashita, D. D. Betts and C. J. Elliott, High-field series expansions and critical properties for the three-state Potts model, J. Phys. A 12 (1979), 1605-1622.
C. Vohwinkel, Yet another way to obtain low temperature expansions for discrete spin systems, Physics Letters B, 301 (1993), 208-212; arXiv:hep-lat/9211052, 1992. See Table 4: multiply by 3/2 to get this sequence.

Cf. other structures: A007271 (b.c.c. lattice), A057374 (square lattice), A057382 (hexagonal lattice), A057390 (honeycomb net), A057398 (kagome net). Cf. A002929 (Ising model).

a(30) corrected, terms a(34) and beyond added from Guttmann & Enting by Andrey Zabolotskiy, Feb 06 2022

Missing term a(36) inserted, name clarified by Andrey Zabolotskiy, Nov 17 2024

cp's OEIS Frontend

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

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A002928 Magnetization for square lattice.

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A003193 Magnetization for body-centered cubic lattice.

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A007270 Low-temperature series for magnetization in zero-field 3-state Potts model on cubic lattice.

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