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16
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 24 (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 12 (2 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 6 (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.
On Valiant’s holographic algorithms
"... Leslie Valiant recently proposed a theory of holographic algorithms. These novel algorithms achieve exponential speedups for certain computational problems compared to naive algorithms for the same problems. The methodology uses Pfaffians and (planar) perfect matchings as basic computational primit ..."
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Cited by 2 (2 self)
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Leslie Valiant recently proposed a theory of holographic algorithms. These novel algorithms achieve exponential speedups for certain computational problems compared to naive algorithms for the same problems. The methodology uses Pfaffians and (planar) perfect matchings as basic computational primitives, and attempts to create exponential cancellations in computation. In this article we survey this new theory of matchgate computations and holographic algorithms.
Higher Dimensional Aztec Diamonds And A (2 d +2)Vertex Model
, 1997
"... . Motivated by the close relationship between the number of perfect matchings of the Aztec diamond graph introduced in [5] and the free energy of the squareice model, we consider a higher dimensional analog of this phenomenon. For d 1, we construct d uniform hypergraphs which generalize the Azte ..."
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Cited by 1 (1 self)
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. Motivated by the close relationship between the number of perfect matchings of the Aztec diamond graph introduced in [5] and the free energy of the squareice model, we consider a higher dimensional analog of this phenomenon. For d 1, we construct d uniform hypergraphs which generalize the Aztec diamonds and we consider a companion ddimensional statistical model (called the 2 d + 2vertex model) whose free energy is given by the logarithm of the number of perfect matchings of our hypergraphs. We prove that the limit defining the free energy per site of the 2 d + 2vertex model exists and we obtain bounds for it. As a consequence, we obtain an especially good asymptotical approximation for the number of matchings of our hypergraphs. 1. Introduction In [5] there are introduced the Aztec diamond graphs, which can be defined as follows. Consider a (2n+1) \Theta (2n+1) chessboard with black corners. The graph whose vertices are the white squares and whose edges connect precisely...
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.
Fully packed loop models on finite geometries
, 901
"... A fully packed loop (FPL) model on the square lattice is the statistical ensemble of all loop configurations, where loops are drawn on the bonds of the lattice, and each loop visits every site once [4,18]. On finite geometries, loops either connect external terminals on the boundary, or form closed ..."
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A fully packed loop (FPL) model on the square lattice is the statistical ensemble of all loop configurations, where loops are drawn on the bonds of the lattice, and each loop visits every site once [4,18]. On finite geometries, loops either connect external terminals on the boundary, or form closed circuits, see for example Figure 1. In this chapter we shall be mainly concerned with FPL models on squares and rectangles with an alternating boundary condition where every other boundary terminal is covered by a loop segment, see Figure 1. Fig. 1. Fully packed loops inside a square with alternating boundary condition. 2 Jan de Gier An FPL model thus describes the statistics of closely packed polygons on a finite geometry. Polygons may be nested, corresponding to punctures studied in Chapter 8. FPL models can be generalised to include weights. In particular we will study FPL models where a weight τ is given to each straight local loop segment. The partition function of an FPL model on various geometries can be computed exactly using its relation to the solvable sixvertex lattice
Fractal structure of a solvable lattice model
, 2009
"... Fractal structure of the sixvertex model is introduced with the use of the IFS (Iterated Function Systems). The fractal dimension satisfies an equation written by the free energy of the sixvertex model. It is pointed out that the transfer matrix method and the nequivalence relation introduced in ..."
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Fractal structure of the sixvertex model is introduced with the use of the IFS (Iterated Function Systems). The fractal dimension satisfies an equation written by the free energy of the sixvertex model. It is pointed out that the transfer matrix method and the nequivalence relation introduced in lattice theories have also been introduced in the area of fractal geometry. All the results can be generalized for the models suitable to the transfer matrix treatment, and hence this gives general relation between solvable lattice models and fractal geometry.