Topology of Gauge Fields and Condensed Matter

Monastyrsky, Michael

doi:10.1007/978-1-4899-2403-2

Cited by 66 publications

(41 citation statements)

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“…The nontrivial aspects of topology and geometry have numerous applications in the modern condensed matter physics where they play an important role in various technologically and physically interesting systems. [4,5] Carbon is undoubtedly a unique element for its structural diversity and exotic geometries and more so in curvature and topology aspects, namely the spherical fullerenes or stable sp 2 C spherical carbon cages, [6,7] spheroidal hyper-/hypofullerenes, [8] cylindrical nanotubes, [9] conical nanocarbons [10 -14] and toroidal nanorings [15] that have attracted a great deal of attention both experimentally and theoretically and opened a new research area in materials science. [16,17] These structures can be collectively referred to as the topologically distinct nanoscale geometric allotropes of carbon.…”

Section: Introduction Topology and Nanocarbonsmentioning

confidence: 99%

Nanocarbon materials: probing the curvature and topology effects using phonon spectra

Gupta

Saxena

2009

J Raman Spectroscopy

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Much has been learned from the use of resonance Raman spectroscopy and high-resolution transmission electron microscopy techniques about the micro-/nanoscopic structure of various nanostructured carbons. However, they still possess some features that are not entirely understood particularly in terms of topological characteristics, which go beyond making a distinction with just the geometrical structure at nanoscale. To effectively utilize the potential of these materials for technological needs, understanding both the geometrical and topological structure and perhaps relating these attributes to physical (optical/electronic, lattice vibrational) properties become indispensable. Here, we make an attempt to describe the differences between various nanostructures and provide geometrical and topological property assessment semiquantitatively by monitoring the phonon spectra using resonance Raman spectroscopy thereby also capturing the electronic spectra. We elucidate the notion of global topology and curvature for a range of technologically important nanoscale carbons including tubular (single-, double-and multiwalled nanotubes, peapod), spherical (hypo-and hyperfullerenes, onion-like carbon) and complex (nanocones, nanohorns, nanodisks and nanorings) geometries. To demonstrate the proof-of-concept, we determined the variation in the prominent Raman bands of the respective materials, represented as D, G and D * (the overtone of D) bands, as a possible topological or curvature trend due to their sensitivity toward structural modification. The latter arises from local topological defects such as pentagons giving rise to curved nanocarbons. In this study, we provide systematics of their variation with respect to their geometric forms and compare with highly oriented pyrolytic graphite and monolayer graphene since the nanocarbons discussed are their derivatives. Once established, this knowledge will provide a powerful machinery to understand newer nanocarbons and indeed point to an unprecedented emergent paradigm of global topology/curvature → property → functionality relationship. We emphasize that these concepts are applicable to other topologically distinct nanomaterials, which include boron-nitride (BN) nanotubes and nanotori, helical gold nanotubes and Möbius conjugated organics. Copyright

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Section: Introduction Topology and Nanocarbonsmentioning

confidence: 99%

Nanocarbon materials: probing the curvature and topology effects using phonon spectra

Gupta

Saxena

2009

J Raman Spectroscopy

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“…Small perturbations usually do not create new PS points or eliminate existing ones, since PS points are individually topo- (16) logically stable [28], [32], but they can cause the changes of the positions of PSs. In this section, we study the stability of PSs to noise addition.…”

Section: Stability Of Pss To Noisementioning

confidence: 99%

“…We can insert (28) and (29) into (30) to get an explicit form. Note that with only the first order of (28) and (29), the probability of (30) reduces to a Gaussian distribution of .…”

Section: Stability Of Pss To Noisementioning

confidence: 99%

“…The image scaling problem can be further solved in a solid framework through key PS selection in Section III-C. Small noise addition or image distortion can change the positions of PS points. However, they are topologically-stable points, and small perturbation usually cannot eliminate or create PS points [28]. We analyze the stability of PS points to noise addition in Section II-C.…”

Section: Phase Singularitymentioning

confidence: 99%

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A Theory of Phase Singularities for Image Representation and its Applications to Object Tracking and Image Matching

Qiao

Wang

Minematsu

et al. 2009

IEEE Trans. on Image Process.

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This paper studies phase singularities (PSs) for image representation. We show that PSs calculated with Laguerre-Gauss filters contain important information and provide a useful tool for image analysis. PSs are invariant to image translation and rotation. We introduce several invariant features to characterize the core structures around PSs and analyze the stability of PSs to noise addition and scale change. We also study the characteristics of PSs in a scale space, which lead to a method to select key scales along phase singularity curves. We demonstrate two applications of PSs: object tracking and image matching. In object tracking, we use the iterative closest point algorithm to determine the correspondences of PSs between two adjacent frames. The use of PSs allows us to precisely determine the motions of tracked objects. In image matching, we combine PSs and scale-invariant feature transform (SIFT) descriptor to deal with the variations between two images and examine the proposed method on a benchmark database. The results indicate that our method can find more correct matching pairs with higher repeatability rates than some well-known methods.

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“…The symmetry group G of the order parameter does not coincide with SO (3), because it is determined by the form of the system potential energy V (Q). The set of values M of the order parameter (the variation range) minimizing the potential V (Q) is called the space of internal states; in this case, M ∼ SO(3)/G, where G is the symmetry group of the order parameter (the isotropy group) [6]. All the points of the set M are equivalent in the sense that each of them can be obtained from another by an SO (3) transformation.…”

Section: Breaking the So(3) Symmetry Of The Lagrangianmentioning

confidence: 99%

Some details of the description of a disordered condensed system using the theory of defect states of orientational order

Vasin

2009

Theor Math Phys

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Using the theory of defect states of orientational order, we describe a disordered condensed system as an elastic medium with linear topological singularities. We show that elastic stress fields produced by linear disclinations are Abelian. In the quasistationary linear approximation, we obtain expressions for linear dislocation and disclination tensor potentials. We show that using the theory of defect states of orientational order, we can describe the α and β relaxations in a supercooled liquid as relaxation processes in the respective disclination and dislocation subsystems.

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Topology of Gauge Fields and Condensed Matter

Cited by 66 publications

References 0 publications

Nanocarbon materials: probing the curvature and topology effects using phonon spectra

Nanocarbon materials: probing the curvature and topology effects using phonon spectra

A Theory of Phase Singularities for Image Representation and its Applications to Object Tracking and Image Matching

Some details of the description of a disordered condensed system using the theory of defect states of orientational order

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