Lattice Permutations and Poisson-Dirichlet Distribution of Cycle Lengths

Großkinsky, Stefan; Lovisolo, Alexander A.; Ueltschi, Daniel

doi:10.1007/s10955-012-0450-9

Cited by 27 publications

(49 citation statements)

References 30 publications

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“…Models on the complete graph are easier to analyse than the corresponding models on a lattice Z d , d ≥ 3; but the heuristics for the latter models is remarkably similar to the one for the former models; see [11,32]. The ideas sketched here are confirmed by the results of numerical simulations of various loop soups, including lattice permutations [12], loop O(N)-models [21], and the random interchange model [4].…”

Section: Random Loop Representationssupporting

confidence: 55%

Quantum Spins and Random Loops on the Complete Graph

Björnberg

Fröhlich

Ueltschi

2019

Commun. Math. Phys.

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We present a systematic analysis of quantum Heisenberg-, xy-and interchange models on the complete graph. These models exhibit phase transitions accompanied by spontaneous symmetry breaking, which we study by calculating the generating function of expectations of powers of the averaged spin density. Various critical exponents are determined. Certain objects of the associated loop models are shown to have properties of Poisson-Dirichlet distributions.

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Section: Random Loop Representationssupporting

confidence: 55%

Quantum Spins and Random Loops on the Complete Graph

Björnberg

Fröhlich

Ueltschi

2019

Commun. Math. Phys.

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“…The outcome is shown in Fig. 12: we (15) and A = 1.58 (20). Again we obtain good overlap of data from different system sizes.…”

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confidence: 59%

Phase transitions in three-dimensional loop models and theCPn−1sigma model

et al. 2013

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We consider the statistical mechanics of a class of models involving close-packed loops with fugacity n on three-dimensional lattices. The models exhibit phases of two types as a coupling constant is varied: in one, all loops are finite, and in the other, some loops are infinitely extended. We show that the loop models are discretisations of CP n−1 σ models. The finite and infinite loop phases represent, respectively, disordered and ordered phases of the σ model, and we discuss the relationship between loop properties and σ model correlators. On large scales, loops are Brownian in an ordered phase and have a non-trivial fractal dimension at a critical point. We simulate the models, finding continuous transitions between the two phases for n = 1, 2, 3 and first order transitions for n ≥ 4. We also give a renormalisation group treatment of the CP n−1 model that shows how a continuous transition can survive for values of n larger than (but close to) two, despite the presence of a cubic invariant in the Landau-Ginzburg description. The results we obtain are of broader relevance to a variety of problems, including SU (n) quantum magnets in (2+1) dimensions, Anderson localisation in symmetry class C, and the statistics of random curves in three dimensions.

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“…This behavior is actually fairly general and concerns many systems where one-dimensional objects have macroscopic size. The distribution PD(1) has been confirmed numerically in a model of lattice permutations [24] and analytically in the related annealed model [25]. More general PD(θ) have been confirmed numerically in O(n) loop models [14].…”

Section: A Macroscopic Loops and Random Partitionsmentioning

confidence: 57%

Ferromagnetism, antiferromagnetism, and the curious nematic phase ofS=1quantum spin systems

Ueltschi

2015

Phys. Rev. E

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Original citation: Ueltschi, Daniel. (2015) Ferromagnetism, antiferromagnetism, and the curious nematic phase ofS=1 quantum spin systems. Physical Review E, 91 (4). 042132. Permanent WRAP URL:http://wrap.warwick.ac.uk/80071 Copyright and reuse:The Warwick Research Archive Portal (WRAP) makes this work by researchers of the University of Warwick available open access under the following conditions. Copyright © and all moral rights to the version of the paper presented here belong to the individual author(s) and/or other copyright owners. To the extent reasonable and practicable the material made available in WRAP has been checked for eligibility before being made available.Copies of full items can be used for personal research or study, educational, or not-for-profit purposes without prior permission or charge. Provided that the authors, title and full bibliographic details are credited, a hyperlink and/or URL is given for the original metadata page and the content is not changed in any way. Publisher statement: © 2015 American Physical Society A note on versions:The version presented here may differ from the published version or, version of record, if you wish to cite this item you are advised to consult the publisher's version. Please see the 'permanent WRAP URL' above for details on accessing the published version and note that access may require a subscription. Ferromagnetism, antiferromagnetism, and the curious nematic phase of S = 1 quantum spin systems Daniel UeltschiDepartment of Mathematics, University of Warwick, Coventry, CV4 7AL, United Kingdom *We investigate the phase diagram of S = 1 quantum spin systems with SU(2)-invariant interactions, at low temperatures and in three spatial dimensions. Symmetry breaking and the nature of extremal states can be studied using random loop representations. The latter confirm the occurrence of ferro-and antiferromagnetic transitions and the breaking of SU(3) invariance. And they reveal the peculiar nature of the nematic extremal states which minimize x (S i x ) 2 .

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Lattice Permutations and Poisson-Dirichlet Distribution of Cycle Lengths

Cited by 27 publications

References 30 publications

Quantum Spins and Random Loops on the Complete Graph

Quantum Spins and Random Loops on the Complete Graph

Phase transitions in three-dimensional loop models and theCPn−1sigma model

Ferromagnetism, antiferromagnetism, and the curious nematic phase ofS=1quantum spin systems

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