Geometric theory of defects

Katanaev, M. O.

doi:10.1070/pu2005v048n07abeh002027

Cited by 96 publications

(74 citation statements)

References 83 publications

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“…This is due to the definition of the triad field in the geometric theory of defects [8]. The resulting metric has discontinuous angular component…”

Section: Pos(cncfg2010)022mentioning

confidence: 99%

“…The net result is the δ ′ -function in the right hand side. At the same time, existence of a dislocation leads to the appearance of nontrivial torsion in space [3][4][5][6][7] (for review, see [8]). This fact is well known in General Relativity which can be equivalently reformulated in a space-time with absolute parallelism (zero curvature) but nontrivial torsion.…”

Section: Introductionmentioning

confidence: 99%

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Torsion and Burgers vector of a tube dislocation

Katanaev¹

2011

Proceedings of Corfu Summer Institute on Elementary Particles and Physics - Workshop on Non Commutative Field Theory and Gravit

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“…This is due to the definition of the triad field in the geometric theory of defects [8]. The resulting metric has discontinuous angular component…”

Section: Pos(cncfg2010)022mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Torsion and Burgers vector of a tube dislocation

Katanaev¹

2011

Proceedings of Corfu Summer Institute on Elementary Particles and Physics - Workshop on Non Commutative Field Theory and Gravit

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“…We explore some of the properties of these configurations in gravity models with non-vanishing curvature, and discuss the possible existence of such monopoles in condensed matter systems. To conclude, we show how the monopoles can be thought of as a natural generalization of the Cartan spiral staircase.The analogy between geometry and defects in gravity theories and in theories of elasticity in solids is an old and well developed field of study [1][2][3][4][5]. Disclinations and dislocations in crystals are defects in the ordered lattice which carry finite curvature and torsion respectively.…”

mentioning

confidence: 99%

“…The analogy between geometry and defects in gravity theories and in theories of elasticity in solids is an old and well developed field of study [1][2][3][4][5]. Disclinations and dislocations in crystals are defects in the ordered lattice which carry finite curvature and torsion respectively.…”

mentioning

confidence: 99%

“…The classical geometric theory of dislocations and disclinations has the Euclidean Poincaré group G = SO(3) T (3) as a gauge group where T (3) is the set of three dimensional translations [9]. Disclinations are defects associated with the rotational degrees of freedom, and in this model, the absence of disclinations is associated with the vanishing of the curvature of the spin connection R ω = 0 [1][2][3][4][5]. Thus, the TM, which can exist in geometries with vanishing curvature is not composed of disclinations.…”

mentioning

confidence: 99%

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Torsional Monopoles and Torqued Geometries in Gravity and Condensed Matter

Randono¹,

Hughes²

2011

Phys. Rev. Lett.

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Torsional degrees of freedom play an important role in modern gravity theories as well as in condensed matter systems where they can be modeled by defects in solids. Here we isolate a class of torsion models that support torsion configurations with a localized, conserved charge that adopts integer values. The charge is topological in nature and the torsional configurations can be thought of as torsional 'monopole' solutions. We explore some of the properties of these configurations in gravity models with non-vanishing curvature, and discuss the possible existence of such monopoles in condensed matter systems. To conclude, we show how the monopoles can be thought of as a natural generalization of the Cartan spiral staircase.The analogy between geometry and defects in gravity theories and in theories of elasticity in solids is an old and well developed field of study [1][2][3][4][5]. Disclinations and dislocations in crystals are defects in the ordered lattice which carry finite curvature and torsion respectively. Transporting a particle around a disclination (dislocation) produces a non-zero rotation (translation) by the end of the cycle. Dislocations are particularly interesting because, while sources of curvature are ubiquitous in the natural universe, the effects of torsion in a gravitational context are thus far negligible experimentally [6]. In solids, however, dislocations affect many important material properties and are present even in the cleanest materials. Thus condensed matter systems can provide useful laboratories for the study of torsion.As we will show, the defects that we will describe cannot be described by the classical geometric theory of elasticity. Instead, these defects may occur in materials described by micropolar elasticity theory [7]. Micropolar elasticity theory (or Cosserat elasticity) is a simple extension of classical elasticity to include local orientational degrees of freedom of the constituent particles/molecules of the elastic medium. The defect we investigate, which we dub a torsional 'monopole' (TM), is a defect that does not require a lattice deformation, but a deformation texture in the local rotational degrees of freedom. Such defects could exist in biological or granular systems (two common systems described by micropolar elasticity) and may affect solids with a strong coupling between orbital electronic motion and local spin or orbital degrees of freedom. Although TMs are not very complicated objects, as evidenced by the structure shown in Fig. 1, there are some subtle issues that require a careful treatment. We present a general treatment of these defects in a gravitational context in flat and curved space and then give an explicit construction of a TM and its relation to defects in solids and the "Cartan spiral staircase." We begin with a simple formulation of our construction in flat space. To isolate the purely torsional degrees of freedom in the absence of other geometric constructs, we will begin with three highly constraining FIG. 1. A cross-section through the o...

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Electron on a cylinder with topological defects in a homogeneous magnetic field

Filgueiras

Oliveira

2011

Annalen der Physik

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In this work we study the effects of the geometry and topology of a cylinder on the energy levels of an electron moving in a homogeneous magnetic field. We consider the existence of topological defects as a screw dislocation and a disclination. When we take the region of movement as the full cylindrical surface, we find that, by increasing the strength of the screw dislocation, the dispersion on the electronic energy levels is affected and monotonically increasing. For an electron moving in an almost flat region we show that the dispersion on the Landau levels decrease monotonically as we increase the strength of the screw dislocation. The lowest Landau level can reach a zero value, leaving the energy of the system solely given by the geometry of the cylinder, which does not depend on the magnetic field. In both situations, as we change the deficit angle of the disclination, we observe that the energy levels are shifted and the magnitude of such shift depends on the magnetic field. The Landau levels for a flat sample are recovered in the limit of an infinite cylinder radius.

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Geometric theory of defects

Cited by 96 publications

References 83 publications

Torsion and Burgers vector of a tube dislocation

Torsion and Burgers vector of a tube dislocation

Torsional Monopoles and Torqued Geometries in Gravity and Condensed Matter

Electron on a cylinder with topological defects in a homogeneous magnetic field

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