A002912 -id:A002912 - cp's OEIS frontend

A002927 Low temperature series for spin-1/2 Ising magnetic susceptibility on 2D square lattice.

0, 0, 1, 8, 60, 416, 2791, 18296, 118016, 752008, 4746341, 29727472, 185016612, 1145415208, 7059265827, 43338407712, 265168691392, 1617656173824, 9842665771649, 59748291677832, 361933688520940, 2188328005246304, 13208464812265559, 79600379336505560, 479025509574159232

Offset: 0

N. J. A. Sloane, Simon Plouffe

nonn

The zero-field susceptibility per spin is 4m^2/kT * Sum_{n >= 0} a(n) * u^n, where u = exp(-4J/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 r = u/(1-u)^2 and dividing by 4. - 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. 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).

Andrey Zabolotskiy, Table of n, a(n) for n = 0..1305
R. J. Baxter and I. G. Enting, Series expansions for corner transfer matrices: the square lattice Ising model, J. Stat. Physics 21 (1979) 103-123.
C. Domb, Ising model, Phase Transitions and Critical Phenomena 3 (1974), 257, 380-381, 384-387, 390-391, 412-423. (Annotated scanned copy)
I. G. Enting, A, J. Guttmann and I. Jensen, Low-Temperature Series Expansions for the Spin-1 Ising Model, arXiv:hep-lat/9410005, 1994; J. Phys. A. 27 (1994) 6987-7005.
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]
Tony Guttmann, Homepage. See Numerical Data, Ising square lattice susceptibility series, Low temperature series.
Iwan Jensen, Series for the Ising model

Cf. A002906 (high-temperature), A002979 (antiferromagnetic susceptibility), A029872 (specific heat), A002928 (magnetization), A002890 (partition function), A047709 (hexagonal lattice), A002912 (honeycomb), A002926 (cubic lattice), A010115 (spin-1 Ising).

a(n) ~ c * n^(3/4) * (1 + sqrt(2))^(2*n), where c = 0.0187325517235678... - Vaclav Kotesovec, May 06 2024

Corrections and updates from Steven Finch

a(0) = a(1) = 0 prepended, terms a(20) and beyond added by Andrey Zabolotskiy, Feb 10 2022

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

Original entry on oeis.org

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

A002978 Low-temperature series in y = exp(2J/kT) for antiferromagnetic susceptibility for the Ising model on honeycomb structure.

Original entry on oeis.org

0, 0, 4, 0, 12, 8, 48, 96, 320, 888, 2748, 8384, 26340, 83568, 268864, 873648, 2865216, 9470784, 31525524, 105594912, 355673804, 1204059144, 4094727168, 13983145888, 47932777680, 164881688088, 568990371212, 1969356192624, 6834965581764, 23782468159920

Offset: 1

N. J. A. Sloane

Previous name was: Susceptibility series for honeycomb.

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

Y. Chan, A. J. Guttmann, B. G. Nickel, 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 641 term in the file Honeycomb_y641_af_.txt.
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. 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.

Cf. A002910, A002912, A007214-A007218, A057391, A057395, A047709, A128834.

From Andrey Zabolotskiy, Mar 03 2021: (Start)
a(n) = 4*A007214(n-3).
G.f.: 8*t(u(y)) - 4*h(y), where t(u) is the g.f. of A047709, h(y) is the g.f. of A002912, and u(y) = y/(1-y+y^2) [Sykes & Fisher, p. 934-935]. (End)

New name from and more terms from Chan et al added by Andrey Zabolotskiy, Mar 03 2021

cp's OEIS Frontend

A002927 Low temperature series for spin-1/2 Ising magnetic susceptibility on 2D square lattice.

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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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A002978 Low-temperature series in y = exp(2J/kT) for antiferromagnetic susceptibility for the Ising model on honeycomb structure.

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