Periodic Striped Ground States in Ising Models with Competing Interactions

Giuliani, Alessandro; Seiringer, Robert

doi:10.1007/s00220-016-2665-0

Cited by 31 publications

(63 citation statements)

References 41 publications

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“…In the continuous setting, to our knowledge this is the first example of a model with local/nonlocal terms in competition such that the functional is invariant under permutation of coordinates and the minimizers display a pattern formation which is one-dimensional. Such behaviour for a smaller range of exponents in the discrete setting was already shown in [1].where J is a positive constant, Per 1 is the 1-perimeter and | · | 1 is the 1-norm and K 1 (ζ) := 1 (|ζ|1+1) p . The perimeter term acts as short-range attracting force and the nonlocal term as repulsive long-range force.…”

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confidence: 64%

Pattern formation for a family of models with local/nonlocal interactions

Daneri¹,

Runa

2018

Proc Appl Math and Mech

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We establish pattern formation for a family of discrete and continuous functionals consisting of a perimeter term and a nonlocal term. In particular, we show that for both the continuous and discrete functionals the global minimizers are exact periodic stripes. One striking feature is that the minimizers are invariant under a smaller group of symmetries than the functional itself. In the literature this phenomenon is often referred to as symmetry breaking. In the continuous setting, to our knowledge this is the first example of a model with local/nonlocal terms in competition such that the functional is invariant under permutation of coordinates and the minimizers display a pattern formation which is one-dimensional. Such behaviour for a smaller range of exponents in the discrete setting was already shown in [1].where J is a positive constant, Per 1 is the 1-perimeter and | · | 1 is the 1-norm and K 1 (ζ) := 1 (|ζ|1+1) p . The perimeter term acts as short-range attracting force and the nonlocal term as repulsive long-range force. We consider J slightly below some critical constant J dsc c (resp. J c ) above which the minimizers are trivial. In order to make our problem well-posed we restrict the functional to [0, L) d -periodic sets. Assuming initially that E is composed of stripes, by using the reflection positivity technique one can see that there exists h * J such that periodic stripes of width and distance h * J are minimizers among sets composed of stripes.In the following theorems we show that such periodic stripes are actually global minimizers, as soon as L is an even multiple of h * J . Theorem 1.1 Let d ≥ 1, p ≥ d + 2. Then there exists τ 0 , such that for every 0 < J c − J < τ 0 , one has that for every k ∈ N and L = 2kh * J , the minimizers of F J,L are optimal periodic stripes. Theorem 1.2 Let d ≥ 1, p ≥ d + 2. Then there exists τ 0 , such that for every 0 < J dsc c − J < τ 0 , one has that for every k ∈ N and L = 2kh * J , the minimizers E J of F dsc J,L are optimal periodic stripes. Some of the ingredients of the paper are: a two-scale analysis, a rigidity result and application of the reflection positivity. In [1], the deviation from being a stripe is measured in terms of "angles" and "holes". Then, a lower bound in terms of "angles" and "holes" is shown. In the continuum one would like to find a characterization of nonoptimality in terms of geometric quantities. However, the discrete concepts namely "angle" and "hole" in the continuum are ill-posed. Moreover, a characterization of the geometry cannot reduce to just "angles" and "holes" due to E ⊂ R d and not E ⊂ Z d . For this reason one needs to find a decomposition of the functional into terms that measure in a certain sense how much a minimizer deviates from being a union of stripes. Such quantities have been introduced in [3] and in [2]. The rigidity estimate expresses the penalization for not being a union of stripes.

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supporting

confidence: 64%

Pattern formation for a family of models with local/nonlocal interactions

Daneri¹,

Runa

2018

Proc Appl Math and Mech

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“…A difficulty with our Peierls argument is to control the change in the number of loops when erasing a contour. We find it convenient to condition on the configuration of external contours away from (x, 0); a similar idea was used in [14]. This gives a domain with complicated shape, but this is not a problem.…”

Section: Contour Representationmentioning

confidence: 99%

“…The research reported in this article was completed during the workshop Many-Body Quantum Systems and Effective Theories at the Mathematisches Forschungsinstitut Oberwolfach, September [11][12][13][14][15][16][17] 2016. We thank the organizers and the institute for the great hospitality and the stimulating program.…”

Section: Acknowledgementsmentioning

confidence: 99%

A direct proof of dimerization in a family of SU(n)-invariant quantum spin chains

Nachtergaele

Ueltschi

2017

Lett Math Phys

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“…Any relation of the present work to [9,28] is unclear, since the measures under consideration are different. Different isoperimetric problems exhibiting "crystallization" (or the optimality of sets consisting of parallel stripes) have been studied in, for example [7,12,14,15,33], though these studies have typically focused only on n = 2 or n = 3.…”

Section: Other Related Workmentioning

confidence: 99%

“…Different isoperimetric problems exhibiting “crystallization” (or the optimality of sets consisting of parallel stripes) have been studied in, for example , though these studies have typically focused only on n = 2 or n = 3.…”

Section: Introductionmentioning

confidence: 99%

A periodic isoperimetric problem related to the unique games conjecture

Heilman

2019

Random Struct Algorithms

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We prove the endpoint case of a conjecture of Khot and Moshkovitz related to the unique games conjecture, less a small error. Let n ≥ 2. Suppose a subset Ω of n‐dimensional Euclidean space double-struckRn satisfies −Ω = Ωc and Ω + v = Ωc (up to measure zero sets) for every standard basis vector v∈double-struckRn. For any x=false(x1,…,xnfalse)∈double-struckRn and for any q ≥ 1, let false‖xfalse‖qq=false|x1false|q+…+false|xnfalse|q and let γnfalse(xfalse)=false(2πfalse)−nfalse/2e−false‖xfalse‖22false/2 . For any x ∈ ∂Ω, let N(x) denote the exterior normal vector at x such that ‖N(x)‖2 = 1. Let B=false{x∈double-struckRn:sinfalse(πfalse(x1+…+xnfalse)false)≥0false}. Our main result shows that B has the smallest Gaussian surface area among all such subsets Ω, less a small error: ∫∂normalΩγnfalse(xfalse)normaldx≥false(1−6×10−9false)∫∂Bγnfalse(xfalse)normaldx+∫∂normalΩtrue(1−false‖Nfalse(xfalse)false‖1ntrue)γnfalse(xfalse)normaldx. In particular, ∫∂normalΩγnfalse(xfalse)normaldx≥false(1−6×10−9false)∫∂Bγnfalse(xfalse)normaldx. Standard arguments extend these results to a corresponding weak inequality for noise stability. Removing the factor 6 × 10−9 would prove the endpoint case of the Khot‐Moshkovitz conjecture. Lastly, we prove a Euclidean analogue of the Khot and Moshkovitz conjecture. The full conjecture of Khot and Moshkovitz provides strong evidence for the truth of the unique games conjecture, a central conjecture in theoretical computer science that is closely related to the P versus NP problem. So, our results also provide evidence for the truth of the unique games conjecture. Nevertheless, this paper does not prove any case of the unique games conjecture.

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Periodic Striped Ground States in Ising Models with Competing Interactions

Cited by 31 publications

References 41 publications

Pattern formation for a family of models with local/nonlocal interactions

Pattern formation for a family of models with local/nonlocal interactions

A direct proof of dimerization in a family of SU(n)-invariant quantum spin chains

A periodic isoperimetric problem related to the unique games conjecture

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