The anisotropic Ising correlations as elliptic integrals: duality and differential equations

McCoy, Barry M.; Maillard, J.‐M.

doi:10.1088/1751-8113/49/43/434004

Cited by 6 publications

(22 citation statements)

References 38 publications

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“…This section is about the behavior of the Z-invariant Ising model as the elliptic parameter k varies. Section 4.2 exhibits a duality relation in the sense of Kramers and Wannier, see also [11,47]. In Section 4.3 we derive the phase diagram of the model and compare it to that of the Z-invariant spanning forests of [10].…”

Section: Duality and Phase Transition In The Z-invariant Ising Modelmentioning

confidence: 99%

“…Complementary The set of all transformations (47) with ad − bc 1 also forms a group, called the extended modular group. The quantity ad − bc is then named the order of the transformation (47).…”

Section: Name Of the Change Of Modulusmentioning

confidence: 99%

“…In the last part we prove explicit, local expressions for the dimer model on a bipartite graph corresponding to the XOR version of this Z-invariant Ising model. √ 1−k 2 on the dual isoradial graph G * yield the same probability measure on polygon configurations of the graph G. The elliptic parameters k and k * can be interpreted as parametrizing dual temperatures, see Section 4.2 and also [11,47].where Z Ising (G, J) is the normalizing constant known as the Ising partition function.A polygon configuration of G is a subset of edges such that every vertex has even degree; let P(G) denote the set of polygon configurations of G. Then, the high temperature expansion [36,37] of the Ising model partition function gives the following identity: Z Ising (G, J) = 2 |V| e∈E cosh J e P∈P(G) e∈P tanh J e .…”

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The Z-invariant Ising model via dimers

Boutillier¹,

Tilière

Raschel

2018

Probab. Theory Relat. Fields

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The Z-invariant Ising model [3] is defined on an isoradial graph and has coupling constants depending on an elliptic parameter k. When k = 0 the model is critical, and as k varies the whole range of temperatures is covered. In this paper we study the corresponding dimer model on the Fisher graph, thus extending our papers [7,8] to the full Z-invariant case. One of our main results is an explicit, local formula for the inverse of the Kasteleyn operator. Its most remarkable feature is that it is an elliptic generalization of [8]: it involves a local function and the massive discrete exponential function introduced in [10]. This shows in particular that Z-invariance, and not criticality, is at the heart of obtaining local expressions. We then compute asymptotics and deduce an explicit, local expression for a natural Gibbs measure. We prove a local formula for the Ising model free energy. We also prove that this free energy is equal, up to constants, to that of the Z-invariant spanning forests of [10], and deduce that the two models have the same second order phase transition in k. Next, we prove a self-duality relation for this model, extending a result of Baxter to all isoradial graphs. In the last part we prove explicit, local expressions for the dimer model on a bipartite graph corresponding to the XOR version of this Z-invariant Ising model. √ 1−k 2 on the dual isoradial graph G * yield the same probability measure on polygon configurations of the graph G. The elliptic parameters k and k * can be interpreted as parametrizing dual temperatures, see Section 4.2 and also [11,47].where Z Ising (G, J) is the normalizing constant known as the Ising partition function.A polygon configuration of G is a subset of edges such that every vertex has even degree; let P(G) denote the set of polygon configurations of G. Then, the high temperature expansion [36,37] of the Ising model partition function gives the following identity: Z Ising (G, J) = 2 |V| e∈E cosh J e P∈P(G) e∈P tanh J e .

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Section: Duality and Phase Transition In The Z-invariant Ising Modelmentioning

confidence: 99%

Section: Name Of the Change Of Modulusmentioning

confidence: 99%

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confidence: 99%

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The Z-invariant Ising model via dimers

Boutillier¹,

Tilière

Raschel

2018

Probab. Theory Relat. Fields

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“…And for any N , the Kramers-Wannier duality under t → 1/t can also be understood via the associated transformation properties of the hypergeometric functions [or those for E(t) and K(t)] [91].…”

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confidence: 99%

Resurgence, Painlevé equations and conformal blocks

Dunne

2019

J. Phys. A: Math. Theor.

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We discuss some physical consequences of the resurgent structure of Painlevé equations and their related conformal block expansions. The resurgent structure of Painlevé equations is particularly transparent when expressed in terms of physical conformal block expansions of the associated tau functions. Resurgence produces an intricate network of inter-relations; some between expansions around different critical points, others between expansions around different instanton sectors of the expansions about the same critical point, and others between different non-perturbative sectors of associated spectral problems, via the Bethe-gauge and Painlevé-gauge correspondences. Resurgence relations exist both for convergent and divergent expansions, and can be interpreted in terms of the physics of phase transitions. These general features are illustrated with three physical examples: correlators of the 2d Ising model, the partition function of the Gross-Witten-Wadia matrix model, and the full counting statistics of one dimensional fermions, associated with Painlevé VI, Painlevé III and Painlevé V, respectively.

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

Abdelaziz¹,

Maillard²

2017

J. Phys. A: Math. Theor.

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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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The anisotropic Ising correlations as elliptic integrals: duality and differential equations

Cited by 6 publications

References 38 publications

The Z-invariant Ising model via dimers

The Z-invariant Ising model via dimers

Resurgence, Painlevé equations and conformal blocks

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

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