Correlation Inequalities for Classical and Quantum XY Models

Benassi, Costanza; Lees, Benjamin; Ueltschi, Daniel

doi:10.1007/978-3-319-58904-6_2

Cited by 4 publications

(2 citation statements)

References 29 publications

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“…A Griffiths inequality was also proven for the classical Heisenberg (or O(3)-spin) ferromagnet in the case of homogeneous rotations and for four-component spin systems such as the |φ| 4 lattice euclidean field [8,11]. We direct the reader to [5] for an overview of results in these cases. Griffiths inequalities are useful tools for proofs of the existence of infinite volume limit of correlation functions and monotonicity of spontaneous magnetisation and also allowing comparisons of aspects, such as the critical temperature, of models with different spin and/or spatial dimension.…”

Section: Introductionmentioning

confidence: 92%

Griffiths inequalities for the $O(N)$-spin model

Lees¹

2022

Preprint

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Section: Introductionmentioning

confidence: 92%

Griffiths inequalities for the $O(N)$-spin model

Lees¹

2022

Preprint

Self Cite

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“…Many extensions are available in the literature. Recent accounts can be found in the book of Friedli and Velenik [50, Sections 3.6, 3.8 and 3.9] and in the review of Benassi-Lees-Ueltschi [15]. We start our discussion by introducing the spin O(n) model with general non-negative coupling constants.…”

Section: Non-negativity and Monotonicity Of Correlationsmentioning

confidence: 99%

Lectures on the Spin and Loop O(n) Models

Peled

Spinka

2019

Springer Proceedings in Mathematics &Amp; Statistics

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The classical spin O(n) model is a model on a d-dimensional lattice in which a vector on the (n − 1)-dimensional sphere is assigned to every lattice site and the vectors at adjacent sites interact ferromagnetically via their inner product. Special cases include the Ising model (n = 1), the XY model (n = 2) and the Heisenberg model (n = 3). We discuss questions of longrange order (spontaneous magnetization) and decay of correlations in the spin O(n) model for different combinations of the lattice dimension d and the number of spin components n. Among the topics presented are the Mermin-Wagner theorem, the Berezinskii-Kosterlitz-Thouless transition, the infra-red bound and Polyakov's conjecture on the two-dimensional Heisenberg model. 2The loop O(n) model is a model for a random configuration of disjoint loops. In these notes we discuss its properties on the hexagonal lattice. The model is parameterized by a loop weight n ≥ 0 and an edge weight x ≥ 0. Special cases include self-avoiding walk (n = 0), the Ising model (n = 1), critical percolation (n = x = 1), dimer model (n = 1, x = ∞), proper 4-coloring (n = 2, x = ∞), integer-valued (n = 2) and tree-valued (integer n >= 3) Lipschitz functions and the hard hexagon model (n = ∞). The object of study in the model is the typical structure of loops. We will review the connection of the model with the spin O(n) model and discuss its conjectured phase diagram, emphasizing the many open problems remaining. We then elaborate on recent results for the self-avoiding walk case and for large values of n.

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An Elementary Proof of Phase Transition in the Planar XY Model

Engelenburg

Lis

2022

Commun. Math. Phys.

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Using elementary methods we obtain a power-law lower bound on the two-point function of the planar XY spin model at low temperatures. This was famously first rigorously obtained by Fröhlich and Spencer (Commun Math Phys 81(4):527–602, 1981) and establishes a Berezinskii–Kosterlitz–Thouless phase transition in the model. Our argument relies on a new loop representation of spin correlations, a recent result of Lammers (Probab Relat Fields, 2021) on delocalisation of general integer-valued height functions, and classical correlation inequalities.

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Correlation Inequalities for Classical and Quantum XY Models

Cited by 4 publications

References 29 publications

Griffiths inequalities for the $O(N)$-spin model

Griffiths inequalities for the $O(N)$-spin model

Lectures on the Spin and Loop O(n) Models

An Elementary Proof of Phase Transition in the Planar XY Model

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