On the Ising Model with Random Boundary Condition

Enter, Aernout van; Netočný, Karel; Schaap, Hendrikjan G.

doi:10.1007/s10955-004-2138-2

Cited by 12 publications

(37 citation statements)

References 47 publications

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“…7 A homomorphism is called an isomorphism if it is bijective. 8 As is well known, H0(N; Z) ∼ = Z ⊕ · · · ⊕ Z for any network N. end in any plaquette p ⊂ N + . Namely the complex Σ forms the (d − 1)-dimensional hypersurfaces which are closed or whose boundaries are in the boundaries ∂N * + of the dual N * + of the unfrustration network N + .…”

Section: Topology Of Unfrustration Networkmentioning

confidence: 98%

“…8 Here Hom(A, B) stands for the set of the homomorphism A −→ B. A proof of (3) is given in Appendix A.…”

Section: Topology Of Unfrustration Networkmentioning

confidence: 99%

“…This suggests that the spins show disorder on the frustration network at finite temperatures. Therefore the boundary effect may behave as random fields [8,9] at the boundary of the unfrustration network. But the domain walls minimize their total size in a ground state.…”

Section: Introductionmentioning

confidence: 99%

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Algebraic topology of spin glasses

Koma

2011

J. Phys. A: Math. Theor.

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We study topology of frustration in d-dimensional Ising spin glasses with d ≥ 2 with nearest-neighbor interactions. We prove the following: For any given spin configuration, the domain walls on the unfrustration network are all transverse to a frustrated loop on the unfrustration network, where a domain wall is defined to be a connected element of the collection of all the (d − 1)-cells which are dual to the bonds having an unfavorable energy, and the unfrustration network is the collection of all the unfrustrated plaquettes. These domain walls are topologically nontrivial because they are all related to the global frustration of a loop on the unfrustration network. Taking account of the thermal stability for the domain walls, we can explain the numerical results that three or higher dimensional systems exhibit a spin glass phase, whereas two-dimensional ones do not. Namely, in two dimensions, the thermal fluctuations of the topologically nontrivial domain walls destroy the order of the frozen spins on the unfrustration network, whereas they do not in three or higher dimensions. This may be interpreted as a global, topological effect of the frustrations.

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Section: Topology Of Unfrustration Networkmentioning

confidence: 98%

“…8 Here Hom(A, B) stands for the set of the homomorphism A −→ B. A proof of (3) is given in Appendix A.…”

Section: Topology Of Unfrustration Networkmentioning

confidence: 99%

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Algebraic topology of spin glasses

Koma

2011

J. Phys. A: Math. Theor.

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“…In order to control the contributions from these contours, we need to perform a multiscale analysis, along the lines of [14], with some large deviation estimates on the probability of these exceptional boundary conditions. We obtain in this way the following Propositions, corresponding to Propositions 3.1-3.2 [10]:…”

Section: Finite Low Temperatures In D = 2 With Strong Boundary Condimentioning

confidence: 88%

“…We will distinguish the ensemble of plus-configurations Ω Λ + in which the spins outside of the exterior contours are plus and similarly the ensemble of minus-configurations Ω Λ − . When a contour ends at the boundary and separates one corner from the rest, this corner is defined to be in the interior, and for contours separating at least two corners from at least two other corners (these are interfaces of some sort) we can make a consistent choice for what we call the interior, see [9,10]. It will turn out that our results do not depend on our precise choice.…”

Section: Finite Low Temperatures With Weak Boundary Conditionsmentioning

confidence: 99%

Incoherent boundary conditions and metastates

Enter

Netočný

Schaap

2006

Institute of Mathematical Statistics Lecture Notes - Monograph Series

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In this contribution we discuss the role which incoherent boundary conditions can play in the study of phase transitions. This is a question of particular relevance for the analysis of disordered systems, and in particular of spin glasses. For the moment our mathematical results only apply to ferromagnetic models which have an exact symmetry between low-temperature phases. We give a survey of these results and discuss possibilities to extend them to some situations where many pure states can coexist. An idea of the proofs as well as the reformulation of our results in the language of Newman-Stein metastates are also presented.Comment: Published at http://dx.doi.org/10.1214/074921706000000176 in the IMS Lecture Notes--Monograph Series (http://www.imstat.org/publications/lecnotes.htm) by the Institute of Mathematical Statistics (http://www.imstat.org

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The Roles of Random Boundary Conditions in Spin Systems

Enter

Endo

2020

Progress in Probability

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On the Ising Model with Random Boundary Condition

Cited by 12 publications

References 47 publications

Algebraic topology of spin glasses

Algebraic topology of spin glasses

Incoherent boundary conditions and metastates

The Roles of Random Boundary Conditions in Spin Systems

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