A002169 -id:A002169 - cp's OEIS frontend

A002165 High temperature series for spin-1/2 Heisenberg specific heat on 3-dimensional f.c.c. lattice.

0, 18, 108, 180, -5040, 162000, 14565600, 563253408, 17544639744, 750412309248, 56646776913408, 4973976625190400, 421817449494804480

Offset: 1

N. J. A. Sloane

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 et al., High-temperature expansions for the spin-1/2 Heisenberg model, Phys. Rev., 164 (1967), 800-817.
C. Domb and D. Wood, On high-temperature expansions for the Heisenberg model, Proc. Physical Soc., 86 (1965), 1-16.
M. G. Gonzalez, B. Bernu, L. Pierre and L. Messio, Finite-temperature phase transitions in S=1/2 three-dimensional Heisenberg magnets from high-temperature series expansions, Phys. Rev. B 107 (2023), 235151; arXiv:2303.03135 [cond-mat.str-el], 2023. See Table V; a_n = a(n)*(-1)^n.
J. Oitmaa and E. Bornilla, High-temperature-series study of the spin-1/2 Heisenberg ferromagnet, Phys. Rev. B, 53 (1996), 14228.
Index entries for sequences related to f.c.c. lattice
Index entries for sequences related to specific heat

Cf. A002169 (cubic), A002167 (b.c.c.); A002166 (susceptibility).

Better description from Steven Finch
a(10) corrected, a(11)-a(12) added from Oitmaa and Bornilla by Andrey Zabolotskiy, Oct 20 2021
a(13) from Gonzalez et al. added by Andrey Zabolotskiy, May 06 2023

A002167 High temperature series for spin-1/2 Heisenberg specific heat on 3-dimensional b.c.c. lattice.

Original entry on oeis.org

0, 12, -24, 168, 1440, 24480, -297024, 28017216, -533681664, 41156316672, -503287538688, 53001415916544, -1839416689004544, 246102905022713856, -9001661201883684864

Offset: 1

N. J. A. Sloane

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 et al., High-temperature expansions for the spin-1/2 Heisenberg model, Phys. Rev., 164 (1967), 800-817.
C. Domb and D. Wood, On high-temperature expansions for the Heisenberg model, Proc. Physical Soc., 86 (1965), 1-16.
M. G. Gonzalez, B. Bernu, L. Pierre and L. Messio, Finite-temperature phase transitions in S=1/2 three-dimensional Heisenberg magnets from high-temperature series expansions, Phys. Rev. B 107 (2023), 235151; arXiv:2303.03135 [cond-mat.str-el], 2023. See Table V; a_n = a(n)*(-1)^n.
J. Oitmaa and E. Bornilla, High-temperature-series study of the spin-1/2 Heisenberg ferromagnet, Phys. Rev. B, 53 (1996), 14228. See Table I and the note added in proof.
Index entries for sequences related to b.c.c. lattice
Index entries for sequences related to specific heat

Cf. A002169 (cubic), A002165 (f.c.c.); A002168 (susceptibility).

Better description from Steven Finch
Terms a(11)-a(14) added from Oitmaa and Bornilla by Andrey Zabolotskiy, Oct 20 2021 and Feb 05 2022
Name clarified, a(15) added from Gonzalez et al. by Andrey Zabolotskiy, May 06 2023

A002170 High temperature series for spin-1/2 Heisenberg susceptibility on 3-dimensional simple cubic lattice.

Original entry on oeis.org

6, 48, 528, 7920, 149856, 3169248, 77046528, 2231209728, 71938507776, 2446325534208, 92886269386752, 3995799894239232, 180512165153832960, 8443006907441565696, 440473891771339603968, 25125124946211876962304, 1444211070518302580146176

Offset: 1

N. J. A. Sloane

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 et al., High-temperature expansions for the spin-1/2 Heisenberg model, Phys. Rev., 164 (1967), 800-817.
C. Domb and D. Wood, On high-temperature expansions for the Heisenberg model, Proc. Physical Soc., 86 (1965), 1-16.
M. G. Gonzalez, B. Bernu, L. Pierre and L. Messio, Finite-temperature phase transitions in S=1/2 three-dimensional Heisenberg magnets from high-temperature series expansions, Phys. Rev. B 107 (2023), 235151; arXiv:2303.03135 [cond-mat.str-el], 2023. See Table VI; b_n = a(n)*(-1)^n/2.
M. D. Kuz'min, Extended high-temperature series for the Spin-1/2 Heisenberg ferromagnet, Phil. Mag. Lett., 99 (2019), 338-350; hal-02390804.

Cf. A002169 (specific heat, or free energy).

Better description from Steven Finch
a(11)-a(14) from Kuz'min added by Andrey Zabolotskiy, Oct 20 2021
a(15)-a(17) from Gonzalez et al. added by Andrey Zabolotskiy, May 10 2023

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

A005402 High temperature series for spin-1/2 Heisenberg specific heat on 2D square lattice.

Original entry on oeis.org

0, 6, -12, -84, 1200, 3120, -249312, 920928, 86274816, -1232035584, -40970012160, 1391730516480, 20983074318336, -1798371774277632, -1850681724997632, 2713439169345073152, -40045819902128750592, -4625352042615223025664, 171173842584165886328832, 8223835353617664214695936

Offset: 1

N. J. A. Sloane

sign

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.
J. Oitmaa and E. Bornilla, High-temperature-series study of the spin-1/2 Heisenberg ferromagnet, Phys. Rev. B, 53 (1996), 14228. See Table I and the note added in proof.
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 Square_20_0.py.
Index entries for sequences related to specific heat

Cf. A005400 (hexagonal), A002169 (cubic); A005401 (susceptibility).

Better description from Steven Finch
a(11)-a(14) added from Oitmaa and Bornilla by Andrey Zabolotskiy, Oct 20 2021 and Feb 05 2022
Terms a(15) and beyond from Pierre, Bernu & Messio added by Andrey Zabolotskiy, Nov 18 2024

A002922 Specific heat for diamond.

Original entry on oeis.org

0, 6, -12, -156, 1680, 21264, -592032, -5349216, 381679872, 1095380736, -384803443200, 2445989918208, 547003852781568

Offset: 1

N. J. A. Sloane

a(n)/(n!*(-4)^n) must be equal to a_n given in Table 2 of the Oitmaa's paper. Domb & Wood's a(1)-a(7) satisfy this relation, a(8) = -5712096 and a(9) = 390388992 do not. The terms in the Data section are unambiguously retrieved from a_n. The next term a(14) is approximately -10793475537844224. The last given digit of a_8 is probably a typo. - Andrey Zabolotskiy, Jun 18 2022

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 and D. Wood, On high-temperature expansions for the Heisenberg model, Proc. Physical Soc., 86 (1965), 1-16.
J. Oitmaa, Diamond lattice Heisenberg antiferromagnet, J. Phys.: Condens. Matter, 30 (2018), 155801.
Index entries for sequences related to specific heat

Cf. A002923, A002165, A002167, A002169.

a(8)-a(9) corrected, a(10)-a(13) added by Andrey Zabolotskiy, Jun 18 2022 using Oitmaa's data

cp's OEIS Frontend

A002165 High temperature series for spin-1/2 Heisenberg specific heat on 3-dimensional f.c.c. lattice.

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A002167 High temperature series for spin-1/2 Heisenberg specific heat on 3-dimensional b.c.c. lattice.

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A002170 High temperature series for spin-1/2 Heisenberg susceptibility on 3-dimensional 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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A005402 High temperature series for spin-1/2 Heisenberg specific heat on 2D square lattice.

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A002922 Specific heat for diamond.

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