Random loop representations for quantum spin systems

Ueltschi, Daniel

doi:10.1063/1.4817865

Cited by 56 publications

(138 citation statements)

References 70 publications

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“…For J 1 close to 1 2 J 2 , the result also holds for d 3 [5,11]. The system has actually Néel order in the direction J 1 = 0, J 2 > 0 [11]. Occurrence of quadrupolar order has recently been proved for J 2 > 0 and small negative J 1 in d 6 [12]; the system should have the stronger Néel order, though.…”

Section: Introductionmentioning

confidence: 77%

“…This strategy has been successfully employed for spin 1 2 quantum Heisenberg and XY models [11] and in loop representations of O(n) models [14]. There is in fact little doubt regarding the nature of extremal states and of symmetry breaking in these models.…”

Section: Introductionmentioning

confidence: 99%

“…The quantum system is very different for two main reasons. First, the direction J 1 = 0, J 2 > 0 is not nematic but it has Néel order [11]; its broken staggered SU(3) invariance is explained in the next subsection. The second reason is that for J 2 > J 1 > 0, extremal states are obtained using the external field + x ( a · S x ) 2 with a "+" sign.…”

Section: The Curious Spin Nematic Phasementioning

confidence: 99%

“…An extension was proposed recently [11] that applies to the spin 1 2 XY model, and to the spin 1 model in the nematic and SU(3) regions 0 J 1 J 2 . We describe it in details in this section.…”

Section: Random Loop Representation For the Nematic And Su(3) Regmentioning

confidence: 99%

“…This expression is actually very interesting and suggests that the spin-spin correlation function has exponential decay in the nematic phase, for all temperatures. The quadrupolar correlation function can also be calculated [11] and it is proportional to the probability that both sites (at time 0) belong to the same loop:…”

Section: Random Loop Representation For the Nematic And Su(3) Regmentioning

confidence: 99%

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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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Section: Introductionmentioning

confidence: 77%

Section: Introductionmentioning

confidence: 99%

Section: The Curious Spin Nematic Phasementioning

confidence: 99%

Section: Random Loop Representation For the Nematic And Su(3) Regmentioning

confidence: 99%

Section: Random Loop Representation For the Nematic And Su(3) Regmentioning

confidence: 99%

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

Ueltschi

2015

Phys. Rev. E

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Uniformly Positive Correlations in the Dimer Model and Macroscopic Interacting Self‐Avoiding Walk in ℤ^d, d ≥ 3

Taggi

2021

Comm Pure Appl Math

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Our first main result is that correlations between monomers in the dimer model in Z d do not decay to 0 when d > 2. This is the first rigorous result about correlations in the dimer model in dimensions greater than 2 and shows that the model behaves drastically differently than in two dimensions, in which case it is integrable and correlations are known to decay to zero polynomially. Such a result is implied by our more general, second main result, which states the occurrence of a phase transition in the model of lattice permutations, which is related to the quantum Bose gas. More precisely, we consider a self-avoiding walk interacting with lattice permutations and we prove that, in the regime of fully packed loops, such a walk is 'long' and the distance between its endpoints grows linearly with the diameter of the box. These results follow from the derivation of a version of the infrared bound from a new general probabilistic settings, with coloured loops and walks interacting at sites and walks entering into the system from some 'virtual' vertices.

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Completely Packed Loop Models

Nahum

2014

Springer Theses

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Random loop representations for quantum spin systems

Cited by 56 publications

References 70 publications

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

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

Uniformly Positive Correlations in the Dimer Model and Macroscopic Interacting Self‐Avoiding Walk in ℤ^d, d ≥ 3

Completely Packed Loop Models

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Random loop representations for quantum spin systems

Cited by 56 publications

References 70 publications

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

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

Uniformly Positive Correlations in the Dimer Model and Macroscopic Interacting Self‐Avoiding Walk in ℤd, d ≥ 3

Completely Packed Loop Models

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Product

Resources

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Uniformly Positive Correlations in the Dimer Model and Macroscopic Interacting Self‐Avoiding Walk in ℤ^d, d ≥ 3