Edge states: topological insulators, superconductors and QCD chiral bags

Asorey, M.; Balachandran, A. P.; Pérez-Pardo, Juan Manuel

doi:10.1007/jhep12(2013)073

Cited by 37 publications

(65 citation statements)

References 22 publications

(27 reference statements)

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“…The theory of chiral boundary conditions is not well developed although many physical applications have been analyzed from this perspective [Ch74], [Rho83], [As13], [As15].…”

Section: Introductionmentioning

confidence: 99%

The topology and geometry of self-adjoint and elliptic boundary conditions for Dirac and Laplace operators

Asorey

Ibort

Marmo

2015

Int. J. Geom. Methods Mod. Phys.

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The theory of self-adjoint extensions of first and second order elliptic differential operators on manifolds with boundary is studied via its most representative instances: Dirac and Laplace operators.The theory is developed by exploiting the geometrical structures attached to them and, by using an adapted Cayley transform on each case, the space M of such extensions is shown to have a canonical group composition law structure.The obtained results are compared with von Neumann's Theorem characterising the self-adjoint extensions of densely defined symmetric operators on Hilbert spaces. The 1D case is thoroughly investigated.The geometry of the submanifold of elliptic self-adjoint extensions M ellip is studied and it is shown that it is a Lagrangian submanifold of the universal Grassmannian Gr. The topology of M ellip is also explored and it is shown that there is a canonical cycle whose dual is the Maslov class of the manifold. Such cycle, called the Cayley surface, plays a relevant role in the study of the phenomena of topology change.Self-adjoint extensions of Laplace operators are discussed in the path integral formalism, identifying a class of them for which both treatments leads to the same results.A theory of dissipative quantum systems is proposed based on this theory and a unitarization theorem for such class of dissipative systems is proved.The theory of self-adjoint extensions with symmetry of Dirac operators is also discussed and a reduction theorem for the self-adjoint elliptic Grasmmannian is obtained.Finally, an interpretation of spontaneous symmetry breaking is offered from the

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“…The theory of chiral boundary conditions is not well developed although many physical applications have been analyzed from this perspective [Ch74], [Rho83], [As13], [As15].…”

Section: Introductionmentioning

confidence: 99%

The topology and geometry of self-adjoint and elliptic boundary conditions for Dirac and Laplace operators

Asorey

Ibort

Marmo

2015

Int. J. Geom. Methods Mod. Phys.

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“…(38)- (40), by adding a homogeneous solution of Eq. (29) in order to fulfill the boundary conditions. In this simple situation, the method of images allows one to readily identify these solutions, and the final configuration can be interpreted in terms of suitable images charges and induced magnetic monopoles.…”

Section: Discussionmentioning

confidence: 99%

“…In the following, we discuss the general solution to Eq. (29). To this end, we require an appropriate adaptation of the standard Green's theorem, from which the solution of Eq.…”

Section: Green's Matrix Methodsmentioning

confidence: 99%

“…(27) and the GF in Eq. (29). Here, the indexes i, j range from 1 to 3, while μ, ν range from 0 to 3.…”

Section: Green's Matrix Methodsmentioning

confidence: 99%

“…In other contexts, θ ED under consideration here has been studied as a 3 þ 1 particular kind of Maxwell-Chern-Simons electrodynamics, where it has received considerable attention [15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31]. And also it has been studied as a restricted subset of the Standard Model extension [32,33], where several results have been achieved, too [34][35][36][37][38][39][40].…”

Section: Introductionmentioning

confidence: 99%

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Electro- and magnetostatics of topological insulators as modeled by planar, spherical, and cylindricalθboundaries: Green’s function approach

2016

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The Green's function method is used to analyze the boundary effects produced by a Chern-Simons extension to electrodynamics. We consider the electromagnetic field coupled to a θ term that is piecewise constant in different regions of space, separated by a common interface Σ, the θ boundary, model which we will refer to as θ electrodynamics. This model provides a correct low-energy effective action for describing topological insulators. Features arising due to the presence of the boundary, such as magnetoelectric effects, are already known in Chern-Simons extended electrodynamics, and solutions for some experimental setups have been found with a specific configuration of sources. In this work we construct the static Green's function in θ electrodynamics for different geometrical configurations of the θ boundary, namely, planar, spherical and cylindrical θ-interfaces. Also, we adapt the standard Green's theorem to include the effects of the θ boundary. These are the most important results of our work, since they allow one to obtain the corresponding static electric and magnetic fields for arbitrary sources and arbitrary boundary conditions in the given geometries. Also, the method provides a well-defined starting point for either analytical or numerical approximations in the cases where the exact analytical calculations are not possible. Explicit solutions for simple cases in each of the aforementioned geometries for θ boundaries are provided. On the one hand, the adapted Green's theorem is illustrated by studying the problem of a pointlike electric charge interacting with a planar topological insulator with prescribed boundary conditions. On the other hand, we calculate the electric and magnetic static fields produced by the following sources: (i) a pointlike electric charge near a spherical θ boundary, (ii) an infinitely straight current-carrying wire near a cylindrical θ boundary and (iii) an infinitely straight uniformly charged wire near a cylindrical θ boundary. Our generalization, when particularized to specific cases, is successfully compared with previously reported results, most of which have been obtained by using the method of images.

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Bulk-Edge Dualities in Topological Matter

Asorey

2019

Springer Proceedings in Physics

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Edge states: topological insulators, superconductors and QCD chiral bags

Cited by 37 publications

References 22 publications

The topology and geometry of self-adjoint and elliptic boundary conditions for Dirac and Laplace operators

The topology and geometry of self-adjoint and elliptic boundary conditions for Dirac and Laplace operators

Electro- and magnetostatics of topological insulators as modeled by planar, spherical, and cylindricalθboundaries: Green’s function approach

Bulk-Edge Dualities in Topological Matter

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