Q-Dependent Susceptibilities in Ferromagnetic Quasiperiodic Z-Invariant Ising Models

Au-Yang, Helen; Perk, Jacques H. H.

doi:10.1007/s10955-006-9213-9

Cited by 11 publications

(31 citation statements)

References 15 publications

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“…Although already present in the work of Kenelly [30], Onsager [51] and Wannier [54], the notion of Z-invariance has been fully developed by Baxter in the context of the integrable Figure 2: A piece of a graph G (plain grey lines) and its dual graph G * (dotted grey lines), and the corresponding bipartite graph G Q (plain black lines). 8-vertex model [3], and then applied to the Ising model and self-dual Potts model [4]; see also [52,2,32]. Z-invariance imposes a strong locality constraint which leads to the parameters of the model satisfying a set of equations known as the Yang-Baxter equations.…”

Section: Z-invariant Ising Model Dimer Models and Massive Laplacianmentioning

confidence: 99%

“…Accordingly, the angles θ e = θ e 2K π have a non-zero imaginary part. We now examine the effect of this reciprocal transformation on the coupling constant J(θ e |k) defined in (2). As it does not seem natural to use complex angles, we extend the formula (2) in the regime k 2 > 1 with only the real parts of the angles.…”

Section: Range Of the Z-invariant Ising Model And Phase Transitionmentioning

confidence: 99%

“…We now examine the effect of this reciprocal transformation on the coupling constant J(θ e |k) defined in (2). As it does not seem natural to use complex angles, we extend the formula (2) in the regime k 2 > 1 with only the real parts of the angles. Using [1, 16.11] in (2), one finds that for k 2 > 1…”

Section: Range Of the Z-invariant Ising Model And Phase Transitionmentioning

confidence: 99%

“…As it does not seem natural to use complex angles, we extend the formula (2) in the regime k 2 > 1 with only the real parts of the angles. Using [1, 16.11] in (2), one finds that for k 2 > 1…”

Section: Range Of the Z-invariant Ising Model And Phase Transitionmentioning

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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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: Z-invariant Ising Model Dimer Models and Massive Laplacianmentioning

confidence: 99%

Section: Range Of the Z-invariant Ising Model And Phase Transitionmentioning

confidence: 99%

Section: Range Of the Z-invariant Ising Model And Phase Transitionmentioning

confidence: 99%

Section: Range Of the Z-invariant Ising Model And Phase Transitionmentioning

confidence: 99%

See 2 more Smart Citations

The Z-invariant Ising model via dimers

Boutillier¹,

Tilière

Raschel

2018

Probab. Theory Relat. Fields

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“…36,37 On the basis of the electronegativity equalization principle, Yang and Wang et al [38][39][40][41][42] designed the Atom-Bond Electronegativity Equalization Method (ABEEM) for large organic and biological molecular charge distribution. Lately, ABEEM model has been fused with MM, i.e., ABEEM/MM, which has been applied to the water systems and ion-water systems [43][44][45][46][47][48] as well as to the conformations of alkane and peptide. [49][50][51] Recently, the ABEEM/MM model has been used to perform dynamics simulations for proteins.…”

Section: Introductionmentioning

confidence: 99%

An estimation method of binding free energy in terms of ABEEMσπ/MM and continuum electrostatics fused into LIE method

2010

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A method is proposed for the estimation of absolute binding free energy of interaction between proteins and ligands. The linear interaction energy method is combined with atom-bond electronegativity equalization method at σπ level Force field (fused into molecular mechanics) and generalized Born continuum model calculation of electrostatic solvation for the estimation of the absolute free energy of binding. The parameters of this method are calibrated by using a training set of 24 HIV-1 protease-inhibitor complexes (PDB entry 1AAQ). A correlation coefficient of 0.93 was obtained with a root mean square deviation of 0.70 kcal mol(-1) . This approach is further tested on seven inhibitor and protease complexes, and it provides small root mean square deviation between the calculated binding free energy and experimental binding free energy without reparametrization. By comparing the radii of gyration and the hydrogen bond distances between ligand and protein of three training model molecules, the consistent comparison result of binding free energy is obtained. It proves that this method of calculating the binding free energy with appropriate structural analysis can be applied to quickly assess new inhibitors of HIV-1 proteases. To test whether the parameters of this method can apply to other drug targets, we have validated this method for the drug target cyclooxygenase-2.

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Statistical Mechanics on Isoradial Graphs

Boutillier

Tilière

2011

Probability in Complex Physical Systems

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Isoradial graphs are a natural generalization of regular graphs which give, for many models of statistical mechanics, the right framework for studying models at criticality. In this survey paper, we first explain how isoradial graphs naturally arise in two approaches used by physicists: transfer matrices and conformal field theory. This leads us to the fact that isoradial graphs provide a natural setting for discrete complex analysis, to which we dedicate one section. Then, we give an overview of explicit results obtained for different models of statistical mechanics defined on such graphs: the critical dimer model when the underlying graph is bipartite, the 2-dimensional critical Ising model, random walk and spanning trees and the q-state Potts model.

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Q-Dependent Susceptibilities in Ferromagnetic Quasiperiodic Z-Invariant Ising Models

Cited by 11 publications

References 15 publications

The Z-invariant Ising model via dimers

The Z-invariant Ising model via dimers

An estimation method of binding free energy in terms of ABEEMσπ/MM and continuum electrostatics fused into LIE method

Statistical Mechanics on Isoradial Graphs

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