The Bethe ansatz for the six-vertex and XXZ models: An exposition

Duminil-Copin, Hugo; Gagnebin, Maxime; Harel, Matan; Manolescu, Ioan; Tassion, Vincent

doi:10.1214/17-ps292

Cited by 15 publications

(29 citation statements)

References 7 publications

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“…While we are mainly interested in it for its connection to the previously discussed models, the six-vertex model is a major object of study on its own right. We do not attempt to give an overview of the six-vertex model here; instead, we refer to [22] and Chapter 8 of [2] (and references therein) for a bibliography on the subject and to the companion paper [8] for details specifically used below.…”

Section: Results For the Six-vertex Modelmentioning

confidence: 99%

“…More precisely, the partition function of a toroidal sixvertex model may be expressed as the trace of the M -th power of a matrix V (depending on N ) called the transfer matrix, which we define next. For more details, see [8].…”

Section: Results For the Six-vertex Modelmentioning

confidence: 99%

“…As discussed in [8], each block V [n] satisfies the assumption of the Perron-Frobenius theorem (1) , and thus has one dominant, positive, simple eigenvalue. For an integer 0 ≤ r ≤ N 2, let Λ r (N ) be the Perron-Frobenius eigenvalue of the block V [N 2−r] , where we emphasize the dependence of Λ r on N (recall that N is even).…”

Section: Results For the Six-vertex Modelmentioning

confidence: 99%

“…It has since been widely studied and developed, with applications in various circumstances, such as the one at hand. Its formulation for the six-vertex model is described in detail in [8]. For completeness, let us briefly discuss this technique again.…”

Section: The Limit Ofmentioning

confidence: 99%

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Continuity of the Phase Transition for Planar Random-Cluster and Potts Models with $${1 \le q \le 4}$$ 1 ≤ q ≤ 4

Duminil-Copin

Sidoravičius

Tassion

2016

Commun. Math. Phys.

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122

218

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We prove that the q-state Potts model and the random-cluster model with cluster weight q > 4 undergo a discontinuous phase transition on the square lattice. More precisely, we show 1. Existence of multiple infinite-volume measures for the critical Potts and random-cluster models, 2. Ordering for the measures with monochromatic (resp. wired) boundary conditions for the critical Potts model (resp. random-cluster model), and 3. Exponential decay of correlations for the measure with free boundary conditions for both the critical Potts and random-cluster models.The proof is based on a rigorous computation of the Perron-Frobenius eigenvalues of the diagonal blocks of the transfer matrix of the six-vertex model, whose ratios are then related to the correlation length of the random-cluster model. As a byproduct, we rigorously compute the correlation lengths of the critical randomcluster and Potts models, and show that they behave as exp(π 2 √ q − 4) as q tends to 4.Section 4: Fourier computations. The study will require certain computations using Fourier decompositions. While these computations are elementary, they may be lengthy, and would break the pace of the proofs. We therefore defer all of them to Section 4.Notation. Most functions hereafter depend on the parameter ∆ = 2−c 2 2 < −1. For ease of notation, we will generally drop the dependency in ∆, and recall it only when it is relevant. We write ∂ i for the partial derivative in the i th coordinate.

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Section: Results For the Six-vertex Modelmentioning

confidence: 99%

Section: Results For the Six-vertex Modelmentioning

confidence: 99%

Section: Results For the Six-vertex Modelmentioning

confidence: 99%

Section: The Limit Ofmentioning

confidence: 99%

See 2 more Smart Citations

Continuity of the Phase Transition for Planar Random-Cluster and Potts Models with $${1 \le q \le 4}$$ 1 ≤ q ≤ 4

Duminil-Copin

Sidoravičius

Tassion

2016

Commun. Math. Phys.

Self Cite

122

218

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“…This part does not invoke probability at all, and relies heavily on exact computations. For more details on the subject, we refer the curious reader to [45,46]. Here, we will only use the following consequence of the study.…”

Section: Discontinuous Phase Transition For the Random-cluster Model mentioning

confidence: 99%

Lectures on the Ising and Potts Models on the Hypercubic Lattice

Duminil-Copin

2019

Springer Proceedings in Mathematics &Amp; Statistics

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Phase transitions are a central theme of statistical mechanics, and of probability more generally. Lattice spin models represent a general paradigm for phase transitions in finite dimensions, describing ferromagnets and even some fluids (lattice gases). It has been understood since the 1980s that random geometric representations, such as the random walk and random current representations, are powerful tools to understand spin models. In addition to techniques intrinsic to spin models, such representations provide access to rich ideas from percolation theory. In recent years, for two-dimensional spin models, these ideas have been further combined with ideas from discrete complex analysis. Spectacular results obtained through these connections include the proofs that interfaces of the two-dimensional Ising model have conformally invariant scaling limits given by SLE curves, that the connective constant of the self-avoiding walk on the hexagonal lattice is given by 2 + √ 2. In higher dimensions, the understanding also progresses with the proof that the phase transition of Potts models is sharp, and that the magnetization of the three-dimensional Ising model vanishes at the critical point. These notes are largely inspired by [39,41,42].

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