A005399 -id:A005399 - cp's OEIS frontend

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

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

A005401 High-temperature series for Heisenberg model susceptibility on square lattice.

Original entry on oeis.org

4, 16, 64, 416, 4544, 23488, -207616, 4205056, 198295552, -2574439424, -112886362112, 3567419838464, 94446596145152, -5636771173998592, -80736001427931136, 11035864514607054848, 15012780903941799936, -25650368909583695740928

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, 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_18_1.py; multiply pol1[1] by 2 to get this sequence.

Cf. A005399 (hexagonal), A002170 (cubic); A005402 (specific heat).

Terms a(11)-a(14) added from Oitmaa and Bornilla by Andrey Zabolotskiy, Oct 20 2021 and Feb 05 2022

Name clarified, terms a(15)-a(18) using Pierre, Bernu & Messio's data added by Andrey Zabolotskiy, Nov 25 2024

A005400 High temperature series for spin-1/2 Heisenberg specific heat on 2D hexagonal lattice.

Original entry on oeis.org

0, 9, 18, -306, -3240, 49176, 1466640, -13626000, -1172668032, 75256704, 1392243773184, 18426692664576, -2213592367094784, -74200148173310976, 4271973657228822528, 294089252618987845632, -8526609981314268364800, -1299100041545138822873088

Offset: 1

N. J. A. Sloane

sign

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_18_0.py.
Index entries for sequences related to specific heat

Cf. A005399 (susceptibility), A005402 (square lattice).

Better description from Steven Finch
a(11)-a(12) added from Oitmaa and Bornilla by Andrey Zabolotskiy, Oct 20 2021
a(13) from Elstner et al. (see table I; signs differ because they consider antiferromagnet, and they mention energy instead of specific heat because the same coefficients are involved, cf. Eqs. (11) and (13) from Oitmaa & Bornilla) added by Andrey Zabolotskiy, Jun 17 2022
a(14)-a(18) from Pierre, Bernu & Messio added by Andrey Zabolotskiy, Nov 25 2024

A002164 E.g.f.: high-temperature series in J/2kT for logarithm of partition function for the spin-1/2 linear (1D) Heisenberg model.

Original entry on oeis.org

0, 3, -6, -30, 360, 504, -44016, 204048, 8261760, -128422272, -1816480512, 76562054400, 124207469568, -51042832542720, 580686719698944, 36632422458820608, -1141184282933624832, -23612862502431719424, 1881307594631033978880, 253019693533000826880

Offset: 1

N. J. A. Sloane

sign

From Andrey Zabolotskiy, Feb 24 2022: (Start)
The power-series parameter may be also written as J/4kT, depending on the particular form of the Hamiltonian.
a(n) = alpha_n / (n * (n-1)), where alpha_n are given in Table I of Shiroishi & Takahashi. (End)

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 = 1..51
G. A. Baker et al., High-temperature series expansions for the spin-1/2 Heisenberg model by the method of irreducible representations of the symmetric group, Phys. Rev., 135 (1964), pages A_1272-A_1277.
Masahiro Shiroishi and Minoru Takahashi, Integral Equation Generates High-Temperature Expansion of the Heisenberg Chain, Phys. Rev. Lett., 89 (2002), 117201.

Cf. A005399.

Name clarified, terms a(14) and beyond added by Andrey Zabolotskiy, Feb 24 2022

cp's OEIS Frontend

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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A005401 High-temperature series for Heisenberg model susceptibility on square lattice.

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A005400 High temperature series for spin-1/2 Heisenberg specific heat on 2D hexagonal lattice.

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A002164 E.g.f.: high-temperature series in J/2kT for logarithm of partition function for the spin-1/2 linear (1D) Heisenberg model.

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