A002891 -id:A002891 - cp's OEIS frontend

A001393 High temperature series for spin-1/2 Ising free energy on 3-dimensional simple cubic lattice.

1, 0, 3, 22, 192, 2046, 24853, 329334, 4649601, 68884356, 1059830112, 16809862992, 273374177222, 4539862959852, 76744615270821, 1317316023432372, 22913901542478978, 403242080061821802, 7169757254509112094, 128654570700129670404, 2327634530912450464791, 42424918919225263486322, 778469235834728913157632, 14371906938404203811137770

Offset: 0

N. J. A. Sloane

z = exp(-f/T) = 2 * cosh(K)^3 * Sum_{n >= 0} a(n) * v^(2*n) where v = tanh(K), K = J/T, T is temperature (in the units of energy), J is the nearest-neighbor interaction, and f is the free energy per spin. See Wipf, pp. 181-182. z is the [geometric average] partition function per spin, so the original name of this entry, "Partition function for cubic lattice", is somewhat more directly related to this sequence. - Andrey Zabolotskiy, Oct 18 2021

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).
Andreas Wipf, Statistical Approach to Quantum Field Theory, LNP 864, Springer, 2013.

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; arXiv:hep-lat/9212032.
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; arXiv:cond-mat/9411002.
G. S. Rushbrooke and J. Eve, High-temperature Ising partition function and related noncrossing polygons for the simple cubic lattice, J. Math. Physics 3 (1962) 185-189. Gives correct a(0)-a(6) and incorrect a(7).
Index entries for sequences related to specific heat

Cf. A001406, A001407, A002891, A010571.

Corrections and updates from Steven Finch

a(14)-a(23) from Andrey Zabolotskiy, Oct 18 2021

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

A002890 Low temperature series for spin-1/2 Ising partition function on 2D square lattice.

Original entry on oeis.org

1, 0, 1, 2, 5, 14, 44, 152, 566, 2234, 9228, 39520, 174271, 787246, 3628992, 17019374, 81011889, 390633382, 1905134695, 9385453576, 46653815395, 233788460256, 1180111379105, 5996452414310, 30653752894948

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).

Andrey Zabolotskiy, Table of n, a(n) for n = 0..500
P. D. Beale, Exact distribution of energies in the two-dimensional Ising model, Phys. Rev. Lett. 76 (1996) 78-81
C. Domb, On the theory of cooperative phenomena in crystals, Advances in Phys., 9 (1960), 149-361.
Steven R. Finch, Lenz-Ising Constants [broken link]
Steven R. Finch, Lenz-Ising Constants [From the Wayback Machine]
G. Siudem, A. Fronczak, and P. Fronczak, Exact low-temperature series expansion for the partition function of the two-dimensional zero-field s= 1/2 Ising model on the infinite square lattice, arXiv preprint arXiv:1410.7963 [math-ph], 2014-2015.
Gandhimohan M. Viswanathan, The hypergeometric series for the partition function of the 2-D Ising model arXiv:1411.2495 [cond-mat.stat-mech], 2014-2015.
Gandhimohan M. Viswanathan, The double hypergeometric series for the partition function of the 2D anisotropic Ising model, arXiv:2104.03430 [cond-mat.stat-mech], 2021.

Cf. A002891.

(* For 25 terms, a PC computation lasts less than half an hour *) m = 48 (* max y exponent *); coes = CoefficientList[ Series[ Log[(1 + y^2)^2 - 2*y*(1 - y^2)*(Cos[2*Pi*u] + Cos[2*Pi*v])], {y, 0, m}], y] // Rest; nint[f_, {n_}] := If[n == 2 || OddQ[n], 0, Print[n]; Integrate[ Integrate[f, {u, 0, 1}], {v, 0, 1}]]; fy = MapIndexed[nint, coes].Table[y^k, {k, 1, m}]; CoefficientList[ Series[ Exp[fy/2], {y, 0, m}] , y^2] (* Jean-François Alcover, Mar 19 2013 *)
CoefficientList[(1+u) Exp[-x HypergeometricPFQ[{1, 1, 3/2, 3/2}, {2, 2, 2}, 16x] /. {x -> (u (1 - u)^2)/(1 + u)^4}] + O[u]^50, u] (* Andrey Zabolotskiy, Feb 12 2022, using the g. f. from Gandhimohan M. Viswanathan, 2014-2015 *)

a(n) ~ exp(2*G/Pi) * (1 + sqrt(2))^(2*n-1) / (Pi*sqrt(2)*n^3), where G is the Catalan's constant A006752. - Vaclav Kotesovec, May 02 2024

Corrections and updates from Steven Finch

"Free energy" changed back to "partition function" (basically the exponential of the free energy) in the name by Andrey Zabolotskiy, Feb 11 2022

A002892 Low-temperature series for partition function for spin-1/2 Ising model on f.c.c. lattice.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 6, -6, 0, 0, 8, 42, -114, 66, 24, 123, 134, -1563, 2262, -405, 846, -2532, -15182, 47961, -37992, 8044, -59694, -57117, 742394, -1233840, 597456, -798392, 1447162, 7898736, -27134598, 27649335

Offset: 0

N. J. A. Sloane

sign

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, On the theory of cooperative phenomena in crystals, Advances in Phys., 9 (1960), 149-361: part 1, part 2.
C. Domb and A. J. Guttmann, Low-temperature series for the Ising model, J. Phys. C: Solid State Phys., 3 (1970), 1652-1660.
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 f.c.c. lattice

Cf. A001407 (high-temperature), A002891 (cubic lattice), A002924 (susceptibility), A003196 (magnetization).

Name clarified, missing minus signs added to a(12) and a(17), and terms a(22) and beyond added by Andrey Zabolotskiy, Feb 14 2022

A030045 Low temperature series for spin-1/2 Ising partition function on 4D simple cubic lattice.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 4, -4, 0, 28, -60, 38, 228, -798, 1036, 1528, -10064, 20266, -432, -116414, 342376, -312232, -1128944, 5218867, -8614080, -6284630, 71744552, -175745087, 80875672, 855493548, -3086044908, 3923353016, 7675455720, -48219069200

Offset: 0

Steven Finch

sign

S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 391-406.
C. Vohwinkel, personal communication.

Steven R. Finch, Lenz-Ising Constants [broken link]
Steven R. Finch, Lenz-Ising Constants [From the Wayback Machine]
M. F. Sykes, Derivation of low-temperature expansions for Ising model, X: The four-dimensional simple hypercubic lattice, J. Phys. A 12 (1979) 879-892.
C. Vohwinkel and P. Weisz, Low-temperature expansion in the d=4 Ising model, Nucl. Phys. B 374 (1992) 647-666. See Table A.1, column f, for the power series which is the logarithm of this power series.

Cf. A002891, A030044, A030046, A030047.

"Free energy" corrected to "partition function" (basically the exponential of the free energy) in the name by Andrey Zabolotskiy, Feb 12 2022

a(33)-a(34) using Vohwinkel & Weisz's data added by Andrey Zabolotskiy, Aug 10 2022

A371049 Low temperature series for spin-1/2 Ising partition function on body-centered cubic lattice.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 4, -4, 0, 28, -60, 44, 204, -750, 1084, 979, -8444, 18886, -7568, -82269, 280288, -348172, -576712, 3677331, -7445964, 569558, 41740944, -126624684

Offset: 1

Andrey Zabolotskiy, Mar 11 2024

The series is in the variable u = exp(-4J/kT).

The expansion of the logarithm of the g.f. of this sequence is given in Domb & Guttmann's Table 1 (with a reference to Sykes et al., 1965) and continued in Eq. (4.14) of Sykes et al., 1973.

Claude Itzykson and Jean-Michel Drouffe, Statistical field theory, vol. 2, Cambridge University Press, 1989. Eq. (120) is supposed to give the logarithm of the g.f., but its second half is erroneously switched with the second half of Eq. (121). These second halves are Eqs. (4.15) and (4.14) of Sykes et al., 1973.

C. Domb and A. J. Guttmann, Low-temperature series for the Ising model, J. Phys. C: Solid State Phys., 3 (1970), 1652-1660.
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. A002891 (simple cubic), A002892 (f.c.c.); A003193 (magnetization), A002925 (ferromagnetic susceptibility), A007218 (antiferromagnetic susceptibility); A001406 (high temperature).

cp's OEIS Frontend

A001393 High temperature series for spin-1/2 Ising free energy on 3-dimensional simple cubic 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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A002890 Low temperature series for spin-1/2 Ising partition function on 2D square lattice.

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A002892 Low-temperature series for partition function for spin-1/2 Ising model on f.c.c. lattice.

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A030045 Low temperature series for spin-1/2 Ising partition function on 4D simple cubic lattice.

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A371049 Low temperature series for spin-1/2 Ising partition function on body-centered cubic lattice.

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