The hypergeometric series for the partition function of the 2D Ising model

Viswanathan, G. M.

doi:10.1088/1742-5468/2015/07/p07004

Cited by 10 publications

(21 citation statements)

References 46 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We see that, in contrast to the low-temperature expansion, a simple rational function like χ (1) H yielding polynomial expressions modulo 2, 4, 8, 16, 32 cannot give rise to the emergence of lacunary series like (35) and (37). For high-temperature series, one must therefore rather ask whether, modulo 2 r , one can distinguish between the full susceptibility χ H and χ (1)…”

Section: The High-temperature Susceptibilitymentioning

confidence: 90%

“…and χ As must be the case, the coefficients are the same up to v 14 for the low-temperature series, and up to v 8 for the high-temperature series, and, beyond, the ratio of the coefficients for χ L (v) and χ (2) L (v) (resp. χ H (v) and χ (1) H (v)) are very close to 1. For the low-temperature series expansion the difference between χ L (v) and χ…”

mentioning

confidence: 81%

“…It is known thatχ (n) = O(w n 2 ), so that the coefficients are the same up to w 14 for the low-temperature series, and up to w 8 for the high-temperature series. Further, one observes that the ratio of the coefficients forχ L andχ (2) L (resp.χ H andχ (1) H ) is very close to 1.…”

mentioning

confidence: 86%

“…Since all the χ (n) 's are diagonals of rational functions [18], similar results are expected for any χ (n) modulo any prime p, and, beyond, modulo any power of a prime p r . As a consequence of the Fermat relations, a p = a, modulo p, one can expect relations, like (1) or (2), to be expressible as functional equations where H(w) p is replaced by H(w p ). Now, the full susceptibility χ is not the diagonal of a rational function, indeed it is not even holonomic [15,17].…”

mentioning

confidence: 99%

“…Now, the full susceptibility χ is not the diagonal of a rational function, indeed it is not even holonomic [15,17]. Therefore, for the full susceptibility, one cannot expect relations like (1) or (2) to exist. Due to the complexity of this function ‡, one might not expect, at first sight, such functional equations for the full susceptibility.…”

mentioning

confidence: 99%

See 4 more Smart Citations

Automata and the susceptibility of the square lattice Ising model modulo powers of primes

Guttmann¹,

Maillard²

2015

J. Phys. A: Math. Theor.

View full text Add to dashboard Cite

We study the full susceptibility of the Ising model modulo powers of primes. We find exact functional equations for the full susceptibility modulo these primes. Revisiting some lesser-known results on discrete finite automata, we show that these results can be seen as a consequence of the fact that, modulo 2 r , one cannot distinguish the full susceptibility from some simple diagonals of rational functions which reduce to algebraic functions modulo 2 r , and, consequently, satisfy exact functional equations modulo 2 r . We sketch a possible physical interpretation of these functional equations modulo 2 r as reductions of a master functional equation corresponding to infinite order symmetries such as the isogenies of elliptic curves. One relevant example is the Landen transformation which can be seen as an exact generator of the Ising model renormalization group. We underline the importance of studying a new class of functions corresponding to ratios of diagonals of rational functions: they reduce to algebraic functions modulo powers of primes and they may have solutions with natural boundaries.

show abstract

Section: The High-temperature Susceptibilitymentioning

confidence: 90%

mentioning

confidence: 81%

mentioning

confidence: 86%

mentioning

confidence: 99%

mentioning

confidence: 99%

See 3 more Smart Citations

Automata and the susceptibility of the square lattice Ising model modulo powers of primes

Guttmann¹,

Maillard²

2015

J. Phys. A: Math. Theor.

View full text Add to dashboard Cite

show abstract

Exact low-temperature series expansion for the partition function of the zero-field Ising model on the infinite square lattice

Siudem

Fronczak

2016

Sci Rep

View full text Add to dashboard Cite

In this paper, we provide the exact expression for the coefficients in the low-temperature series expansion of the partition function of the two-dimensional Ising model on the infinite square lattice. This is equivalent to exact determination of the number of spin configurations at a given energy. With these coefficients, we show that the ferromagnetic–to–paramagnetic phase transition in the square lattice Ising model can be explained through equivalence between the model and the perfect gas of energy clusters model, in which the passage through the critical point is related to the complete change in the thermodynamic preferences on the size of clusters. The combinatorial approach reported in this article is very general and can be easily applied to other lattice models.

show abstract

The double hypergeometric series for the partition function of the 2D anisotropic Ising model

Viswanathan¹

2021

J. Stat. Mech.

Self Cite

View full text Add to dashboard Cite

In 1944 Lars Onsager published the exact partition function of the ferromagnetic Ising model on the infinite square lattice in terms of a definite integral. Only in the literature of the last decade, however, has it been recast in terms of special functions. Until now all known formulas for the partition function in terms of special functions have been restricted to the important special case of the isotropic Ising model with symmetric couplings. Indeed, the anisotropic model is more challenging because there are two couplings and hence two reduced temperatures, one for each of the two axes of the square lattice. Hence, standard special functions of one variable are inadequate to the task. Here, we reformulate the partition function of the anisotropic Ising model in terms of the Kampé de Fériet function, which is a double hypergeometric function in two variables that is more general than the Appell hypergeometric functions. Finally, we present hypergeometric formulas for the generating function of multipolygons of given length on the infinite square lattice, for isotropic as well as anisotropic edge weights. For the isotropic case, the results allow easy calculation, to arbitrary order, of the celebrated series found by Cyril Domb.

show abstract

The hypergeometric series for the partition function of the 2D Ising model

Cited by 10 publications

References 46 publications

Automata and the susceptibility of the square lattice Ising model modulo powers of primes

Automata and the susceptibility of the square lattice Ising model modulo powers of primes

Exact low-temperature series expansion for the partition function of the zero-field Ising model on the infinite square lattice

The double hypergeometric series for the partition function of the 2D anisotropic Ising model

Contact Info

Product

Resources

About