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

Author

N. J. A. Sloane, Simon Plouffe

Keywords

• nonn

Comments

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

References

• 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).

Links

• 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

Crossrefs

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).

Formula

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

Extensions

Corrections and updates from Steven Finch

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