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Symmetry, Volume 5, Issue 4 (December 2013), Pages 271-354

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Research

Open AccessArticle A New Route to the Majorana Equation
Symmetry 2013, 5(4), 271-286; doi:10.3390/sym5040271
Received: 13 May 2013 / Revised: 30 August 2013 / Accepted: 12 September 2013 / Published: 26 September 2013
Cited by 3 | PDF Full-text (289 KB) | HTML Full-text | XML Full-text
Abstract
In this paper, we suggest an alternative strategy to derive the complex two-component Majorana equation with a mass term and elucidate the related Lorentz transformation. The Majorana equation is established completely on its own, rather than derived from the chiral Dirac equation. Thereby,
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In this paper, we suggest an alternative strategy to derive the complex two-component Majorana equation with a mass term and elucidate the related Lorentz transformation. The Majorana equation is established completely on its own, rather than derived from the chiral Dirac equation. Thereby, use is made of the complex conjugation operator and Pauli spin matrices only. The eigenfunctions of the two-component complex Majorana equation are also calculated. The associated quantum fields are found to describe particles and antiparticles, which have opposite mean helicities and are not their own antiparticles, but correspond to two independent degrees of freedom. The four-component real Dirac equation in its Majorana representation is shown to be the natural outcome of the two-component complex Majorana equation. Both types of equations come in two forms, which correspond to the irreducible left- and right-chiral representations of the Lorentz group. Full article
Open AccessArticle Multiple Solutions to Implicit Symmetric Boundary Value Problems for Second Order Ordinary Differential Equations (ODEs): Equivariant Degree Approach
Symmetry 2013, 5(4), 287-312; doi:10.3390/sym5040287
Received: 29 August 2013 / Revised: 24 October 2013 / Accepted: 28 October 2013 / Published: 7 November 2013
Cited by 3 | PDF Full-text (821 KB) | HTML Full-text | XML Full-text
Abstract
In this paper, we develop a general framework for studying Dirichlet Boundary Value Problems (BVP) for second order symmetric implicit differential systems satisfying the Hartman-Nagumo conditions, as well as a certain non-expandability condition. The main result, obtained by means of the equivariant degree
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In this paper, we develop a general framework for studying Dirichlet Boundary Value Problems (BVP) for second order symmetric implicit differential systems satisfying the Hartman-Nagumo conditions, as well as a certain non-expandability condition. The main result, obtained by means of the equivariant degree theory, establishes the existence of multiple solutions together with a complete description of their symmetric properties. The abstract result is supported by a concrete example of an implicit system respecting D4-symmetries. Full article
(This article belongs to the Special Issue Symmetry Breaking)
Open AccessArticle Interplay between Point-Group Symmetries and the Choice of the Bloch Basis in Multiband Models
Symmetry 2013, 5(4), 313-343; doi:10.3390/sym5040313
Received: 8 October 2013 / Revised: 30 October 2013 / Accepted: 4 November 2013 / Published: 11 November 2013
Cited by 2 | PDF Full-text (217 KB) | HTML Full-text | XML Full-text
Abstract
We analyze the point-group symmetries of generic multiband tight-binding models with respect to the transformation properties of the effective interactions. While the vertex functions in the orbital language may transform non-trivially under point-group operations, their point-group behavior in the band language can be
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We analyze the point-group symmetries of generic multiband tight-binding models with respect to the transformation properties of the effective interactions. While the vertex functions in the orbital language may transform non-trivially under point-group operations, their point-group behavior in the band language can be simplified by choosing a suitable Bloch basis. We first give two analytically accessible examples. Then, we show that, for a large class of models, a natural Bloch basis exists, in which the vertex functions in the band language transform trivially under all point-group operations. As a consequence, the point-group symmetries can be used to reduce the computational effort in perturbative many-particle approaches, such as the functional renormalization group. Full article
Open AccessArticle Effect of Symmetry Breaking on Electronic Band Structure: Gap Opening at the High Symmetry Points
Symmetry 2013, 5(4), 344-354; doi:10.3390/sym5040344
Received: 23 September 2013 / Revised: 11 November 2013 / Accepted: 3 December 2013 / Published: 9 December 2013
Cited by 2 | PDF Full-text (837 KB) | HTML Full-text | XML Full-text
Abstract
Some characteristic features of band structures, like the band degeneracy at high symmetry points or the existence of energy gaps, usually reflect the symmetry of the crystal or, more precisely, the symmetry of the wave vector group at the relevant points of the
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Some characteristic features of band structures, like the band degeneracy at high symmetry points or the existence of energy gaps, usually reflect the symmetry of the crystal or, more precisely, the symmetry of the wave vector group at the relevant points of the Brillouin zone. In this paper, we will illustrate this property by considering two-dimensional (2D)-hexagonal lattices characterized by a possible two-fold degenerate band at the K points with a linear dispersion (Dirac points). By combining scanning tunneling spectroscopy and angle-resolved photoemission, we study the electronic properties of a similar system: the Ag/Cu(111) interface reconstruction characterized by a hexagonal superlattice, and we show that the gap opening at the K points of the Brillouin zone of the reconstructed cell is due to the symmetry breaking of the wave vector group. Full article
(This article belongs to the Special Issue Symmetry Breaking)
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