A030044 -id:A030044 - cp's OEIS frontend

A010572 High-temperature coefficients for the internal energy for spin-1/2 Ising model on 4-d cubic lattice.

4, 24, 432, 10512, 290552, 8800432, 284289160, 9621738448

Offset: 0

M. E. Fisher and D. S. Gaunt, Ising model and self-avoiding walks on hypercubical lattices and "high-density" expansions, Phys. Rev. 133 (1964) A224-A239.

Cf. A010571 (3D), A010573 (5D), A010574 (6D), A030044 (partition function), A010557 (fourth-field derivative of free energy), A010563.

Sum_{n>=0} a(n) * v^(2*n+1) = v*q/2 + (1-v^2) * f'(v) / f(v), where f(v) = Sum_{n>=0} A030044(n) * v^(2*n) and q = 8 is the number of nearest neighbors. - Andrey Zabolotskiy, Feb 16 2022

a(5)-a(7) from Andrey Zabolotskiy, Feb 16 2022, corrected Nov 26 2024

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

A030048 High temperature series for spin-1/2 Ising partition function on 5D simple cubic lattice.

Original entry on oeis.org

1, 0, 10, 180, 5075, 180496

Offset: 0

Steven Finch

S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 391-406.

Daniel Andrén, Series expansion for the density of states of the Ising and Potts models, arXiv:0706.3116 [cond-mat.str-el], 2007.
Steven R. Finch, Lenz-Ising Constants [broken link]
Steven R. Finch, Lenz-Ising Constants [From the Wayback Machine]
M. E. Fisher and D. S. Gaunt, Ising model and self-avoiding walks on hypercubical lattices and "high-density" expansions, Phys. Rev. 133 (1964) A224-A239. See p. A226, in particular Eq. (2.11) (together with Eq. (4.10)), for the power series with coefficients g_n, which is the logarithm of this power series, and its physical and combinatorial interpretations.

Cf. A010579 (susceptibility), A010573 (internal energy), A001393 (3D cubic lattice), A030044 (4D cubic lattice).

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

a(5) added by Andrey Zabolotskiy, Jun 30 2022 using Andrén's data (see his Table 4, column b_n^5 for the coefficients of the expansion of the logarithm of the g.f. of this sequence)

A010563 High-temperature expansion for Ising model spin-spin correlation function on 4-d cubic lattice.

Original entry on oeis.org

1, 6, 108, 2628, 72638, 2200108, 71072290, 2405434612

Offset: 0

N. J. A. Sloane

M. A. Moore, Critical behavior of the four-dimensional Ising ferromagnet and the breakdown of scaling, Phys. Rev. B 1 (1970), 2238-2240.

Cf. A010572, A030044 (partition function), A010557 (fourth-field derivative of free energy); A010564 ("HBCC" lattice), A010565 (D_4 lattice).

a(n) = A010572(n) / 4. - Andrey Zabolotskiy, Nov 19 2024

Name clarified and terms a(6)-a(7) using data from A010572 added by Andrey Zabolotskiy, Nov 25 2024

A030049 High temperature series for spin-1/2 Ising partition function on 6D simple cubic lattice.

Original entry on oeis.org

1, 0, 15, 350, 13380, 664566

Offset: 0

Steven Finch

S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 391-406.

Steven R. Finch, Lenz-Ising Constants [broken link]
Steven R. Finch, Lenz-Ising Constants [From the Wayback Machine]
M. E. Fisher and D. S. Gaunt, Ising model and self-avoiding walks on hypercubical lattices and "high-density" expansions, Phys. Rev. 133 (1964) A224-A239.

Cf. A010574 (internal energy), A010580 (susceptibility); A001393 (3D cubic lattice), A030044 (4D), A030048 (5D).

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

a(5) added using data from A010574 by Andrey Zabolotskiy, Jan 20 2023

cp's OEIS Frontend

A010572 High-temperature coefficients for the internal energy for spin-1/2 Ising model on 4-d cubic 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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A030048 High temperature series for spin-1/2 Ising partition function on 5D simple cubic lattice.

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A010563 High-temperature expansion for Ising model spin-spin correlation function on 4-d cubic lattice.

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A030049 High temperature series for spin-1/2 Ising partition function on 6D simple cubic lattice.

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