Experimental mathematics on the magnetic susceptibility of the square lattice Ising model

Boukraa, S.; Guttmann, A J; Hassani, S.; Jensen, Iwan; Maillard, J.‐M.; Nickel, B. G.; Zenine, N.

doi:10.1088/1751-8113/41/45/455202

Cited by 43 publications

(262 citation statements)

References 31 publications

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“…The spin susceptibility is related to the zero momentum value of the Green function by To this order the series expansion agrees with the ones from [34,35]. For a library of high-temperature series expansions, see [36].…”

Section: Magnetic Susceptibilitysupporting

confidence: 62%

“…The first element along and the axis and the diagonal can be recast in terms of complete Figure 9. The magnetic susceptibility versus ζ for different expansion orders from 12 to 1 (top to bottom), compared to the order 100 result-the converged answer over this plotting range-obtained from [34], which shows a divergence in good agreement with the critical exponent…”

Section: Spin-spin Correlation Functionmentioning

confidence: 81%

“…For a library of high-temperature series expansions, see [36]. Currently, the series is known (at least) up to order 2000 and still topic of active research [32,34]. The series is convergent for any finite expansion order, i.e., in the thermodynamic limit the infinite series will diverge first at the phase transition point.…”

Section: Magnetic Susceptibilitymentioning

confidence: 99%

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Grassmannization of classical models

et al. 2016

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Applying Feynman diagrammatics to non-fermionic strongly correlated models with local constraints might seem generically impossible for two separate reasons: (i) the necessity to have a Gaussian (noninteracting) limit on top of which the perturbative diagrammatic expansion is generated by Wick's theorem, and (ii) Dyson's collapse argument implying that the expansion in powers of coupling constant is divergent. We show that for arbitrary classical lattice models both problems can be solved/ circumvented by reformulating the high-temperature expansion (more generally, any discrete representation of the model) in terms of Grassmann integrals. Discrete variables residing on either links, plaquettes, or sites of the lattice are associated with the Grassmann variables in such a way that the partition function (as well as all correlation functions) of the original system and its Grassmannfield counterpart are identical. The expansion of the latter around its Gaussian point generates Feynman diagrams. Our work paves the way for studying lattice gauge theories by treating bosonic and fermionic degrees of freedom on equal footing.

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Section: Magnetic Susceptibilitysupporting

confidence: 62%

Section: Spin-spin Correlation Functionmentioning

confidence: 81%

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Grassmannization of classical models

et al. 2016

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“…The emergence of such generalized covariance (125) for the representation of the Landen transformation (and more generally the modular correspondences providing exact representations of the generators of the renormalization group) on the other nfold integralsχ (n) 's of the susceptibility of the Ising model [32,33,34] is a challenging open problem, that will require one to consider reducible operators (see subsection 4.2).…”

Section: Pullback Symmetry Of An Operator Up To Equivalence Of Operatorsmentioning

confidence: 99%

Schwarzian conditions for linear differential operators with selected differential Galois groups

Abdelaziz¹,

Maillard²

2017

J. Phys. A: Math. Theor.

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Abstract. We show that non-linear Schwarzian differential equations emerging from covariance symmetry conditions imposed on linear differential operators with hypergeometric function solutions can be generalized to arbitrary order linear differential operators with polynomial coefficients having selected differential Galois groups. For order three and order four linear differential operators we show that this pullback invariance up to conjugation eventually reduces to symmetric powers of an underlying order-two operator. We give, precisely, the conditions to have modular correspondences solutions for such Schwarzian differential equations, which was an open question in a previous paper. We analyze in detail a pullbacked hypergeometric example generalizing modular forms, that ushers a pullback invariance up to operator homomorphisms. We expect this new concept to be well-suited in physics and enumerative combinatorics. We finally consider the more general problem of the equivalence of two different orderfour linear differential Calabi-Yau operators up to pullbacks and conjugation, and clarify the cases where they have the same Yukawa couplings.

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“…Setting out to find the precise covariance of some of the physical quantities related to the 2-D Ising model, like the partition function per site, the correlation functions, the n-fold correlations χ (n) associated with the full susceptibility [16,17,18,19], with respect to transformations of the Landen type (1), is a difficult task. An easier goal would be to find a covariance, not on the selected ¶ linear differential operators that † In their pioneering work Julia, Fatou and Ritt the theory of iteration of rational functions was seen as a method for investigating functional equations [1,2,3].…”

Section: Introduction: Infinite Order Symmetriesmentioning

confidence: 99%

Modular forms, Schwarzian conditions, and symmetries of differential equations in physics

Abdelaziz¹,

Maillard²

2017

J. Phys. A: Math. Theor.

Self Cite

110

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Abstract.We give examples of infinite order rational transformations that leave linear differential equations covariant. These examples are non-trivial yet simple enough illustrations of exact representations of the renormalization group. We first illustrate covariance properties on order-two linear differential operators associated with identities relating the same 2 F 1 hypergeometric function with different rational pullbacks. These rational transformations are solutions of a differentially algebraic equation that already emerged in a paper by Casale on the Galoisian envelopes. We provide two new and more general results of the previous covariance by rational functions: a new Heun function example and a higher genus 2 F 1 hypergeometric function example. We then focus on identities relating the same 2 F 1 hypergeometric function with two different algebraic pullback transformations: such remarkable identities correspond to modular forms, the algebraic transformations being solution of another differentially algebraic Schwarzian equation that also emerged in Casale's paper. Further, we show that the first differentially algebraic equation can be seen as a subcase of the last Schwarzian differential condition, the restriction corresponding to a factorization condition of some associated order-two linear differential operator. Finally, we also explore generalizations of these results, for instance, to 3 F 2 , hypergeometric functions, and show that one just reduces to the previous 2 F 1 cases through a Clausen identity. The question of the reduction of these Schwarzian conditions to modular correspondences remains an open question. In a 2 F 1 hypergeometric framework the Schwarzian condition encapsulates all the modular forms and modular equations of the theory of elliptic curves, but these two conditions are actually richer than elliptic curves or 2 F 1 hypergeometric functions, as can be seen on the Heun and higher genus example. This work is a strong incentive to develop more differentially algebraic symmetry analysis in physics.

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Experimental mathematics on the magnetic susceptibility of the square lattice Ising model

Cited by 43 publications

References 31 publications

Grassmannization of classical models

Grassmannization of classical models

Schwarzian conditions for linear differential operators with selected differential Galois groups

Modular forms, Schwarzian conditions, and symmetries of differential equations in physics

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