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Completeness of the Bethe ansatz for the six and eightvertex models
 J. Stat. Phys
"... We discuss some of the difficulties that have been mentioned in the literature in connection with the Bethe ansatz for the sixvertex model and XXZ chain, and for the eightvertex model. In particular we discuss the “beyond the equator”, infinite momenta and exact complete string problems. We show h ..."
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Cited by 31 (1 self)
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We discuss some of the difficulties that have been mentioned in the literature in connection with the Bethe ansatz for the sixvertex model and XXZ chain, and for the eightvertex model. In particular we discuss the “beyond the equator”, infinite momenta and exact complete string problems. We show how they can be overcome and conclude that the coordinate Bethe ansatz does indeed give a complete set of states, as expected.
Exact solution of the sixvertex model with domain wall boundary conditions. Critical line between . . .
, 2008
"... This is a continuation of the papers [4] of Bleher and Fokin and [6] of Bleher and Liechty, in which the large n asymptotics is obtained for the partition function Zn of the sixvertex model with domain wall boundary conditions in the disordered and ferroelectric phases, respectively. In the presen ..."
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Cited by 31 (7 self)
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This is a continuation of the papers [4] of Bleher and Fokin and [6] of Bleher and Liechty, in which the large n asymptotics is obtained for the partition function Zn of the sixvertex model with domain wall boundary conditions in the disordered and ferroelectric phases, respectively. In the present paper we obtain the large n asymptotics of Zn on the critical line between these two phases.
The six and eightvertex models revisited
 J. Stat. Phys
"... Elliott Lieb’s icetype models opened up the whole field of solvable models in statistical mechanics. Here we discuss the “commuting transfer matrix” T,Q equations for these models, writing them in a more explicit and transparent notation that we believe offers new insights. The approach manifests t ..."
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Cited by 10 (0 self)
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Elliott Lieb’s icetype models opened up the whole field of solvable models in statistical mechanics. Here we discuss the “commuting transfer matrix” T,Q equations for these models, writing them in a more explicit and transparent notation that we believe offers new insights. The approach manifests the relationship between the sixvertex and chiral Potts models, and between the eightvertex and KashiwaraMiwa models.
Parafermionic observables and their applications to planar statistical physics models
, 2013
"... This volume is based on the PhD thesis of the author. Through the examples of the selfavoiding walk, the randomcluster model, the Ising model and others, the book explores in details two important techniques: 1. Discrete holomorphicity and parafermionic observables, which have been used in the pa ..."
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Cited by 6 (2 self)
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This volume is based on the PhD thesis of the author. Through the examples of the selfavoiding walk, the randomcluster model, the Ising model and others, the book explores in details two important techniques: 1. Discrete holomorphicity and parafermionic observables, which have been used in the past few years to study planar models of statistical physics (in particular their conformal invariance), such as randomcluster models and loop O(n)models. 2. The RussoSeymourWelsh theory for percolationtype models with dependence. This technique was initially available for Bernoulli percolation only. Recently, it has been extended to models with dependence, thus opening the way to a deeper study of their critical regime. The book is organized as follows. The first part provides a general introduction to planar statistical physics, as well as a first example of the parafermionic observable and its application to the computation of the connective constant for the selfavoiding walk on the hexagonal lattice. The second part deals with the family of randomcluster models. It studies the RussoSeymourWelsh theory of crossing probabilities for these models. As an application, the critical point of the randomcluster model is computed on the square lattice. Then, the parafermionic observable is introduced and two of its applications are described in detail. This part contains a chapter describing basic properties of the randomcluster model. The third part is devoted to the Ising model and its randomcluster representation, the FKIsing model. After a first chapter gathering the basic properties of the Ising model, the theory of sholomorphic functions as well as Smirnov and ChelkakSmirnov’s proofs of conformal invariance (for these two models) are presented. Conformal invariance paves the way to a better understanding of the critical phase and the two next chapters are devoted to the study of the geometry of the critical phase, as well as the relation between the critical and nearcritical phases. The last part presents possible directions of future research by describing other models and several open questions.
Baxter’s relations and spectra of quantum integrable models, Preprint arXiv
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The scaling limit of the energy correlations in non integrable Ising models
, 2012
"... Dedicated to Elliott Lieb on the occasion of his 80th birthday We obtain an explicit expression for the multipoint energy correlations of a non solvable twodimensional Ising models with nearest neighbor ferromagnetic interactions plus a weak finite range interaction of strength λ, in a scaling limi ..."
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Dedicated to Elliott Lieb on the occasion of his 80th birthday We obtain an explicit expression for the multipoint energy correlations of a non solvable twodimensional Ising models with nearest neighbor ferromagnetic interactions plus a weak finite range interaction of strength λ, in a scaling limit in which we send the lattice spacing to zero and the temperature to the critical one. Our analysis is based on an exact mapping of the model into an interacting lattice fermionic theory, which generalizes the one originally used by Schultz, Mattis and Lieb for the nearest neighbor Ising model. The interacting model is then analyzed by a multiscale method first proposed by Pinson and Spencer. If the lattice spacing is finite, then the correlations cannot be computed in closed form: rather, they are expressed in terms of infinite, convergent, power series in λ. In the scaling limit, these infinite expansions radically simplify and reduce to the limiting energy correlations of the integrable Ising model, up to a finite renormalization of the parameters. Explicit bounds on the speed of convergence to the scaling limit are derived.