Emergence of a new symmetry class for Bogoliubov–de Gennes (BdG) Hamiltonians: expanding 10-fold symmetry classes

2020 ◽  
Vol 93 (3) ◽  
pp. 287-300
Author(s):  
Ranjith Kumar R ◽  
Sujit Sarkar
1976 ◽  
Vol 28 (6) ◽  
pp. 1311-1319 ◽  
Author(s):  
L. J. Cummings ◽  
R. W. Robinson

A formula is derived for the dimension of a symmetry class of tensors (over a finite dimensional complex vector space) associated with an arbitrary finite permutation group G and a linear character of x of G. This generalizes a result of the first author [3] which solved the problem in case G is a cyclic group.


2021 ◽  
pp. 108128652110108
Author(s):  
Marc Olive ◽  
Boris Kolev ◽  
Rodrigue Desmorat ◽  
Boris Desmorat

We formulate effective necessary and sufficient conditions to identify the symmetry class of an elasticity tensor, a fourth-order tensor which is the cornerstone of the theory of elasticity and a toy model for linear constitutive laws in physics. The novelty is that these conditions are written using polynomial covariants. As a corollary, we deduce that the symmetry classes are affine algebraic sets, a result which seems to be new. Meanwhile, we have been lead to produce a minimal set of 70 generators for the covariant algebra of a fourth-order harmonic tensor and introduce an original generalized cross-product on totally symmetric tensors. Finally, using these tensorial covariants, we produce a new minimal set of 294 generators for the invariant algebra of the elasticity tensor.


2020 ◽  
Vol 141 (2) ◽  
pp. 349-361
Author(s):  
Oliver Stahn ◽  
Wolfgang H. Müller ◽  
Albrecht Bertram

Abstract For a given elastic stiffness tetrad an algorithm is provided to determine the distance of this particular tetrad to all tetrads of a prescribed symmetry class. If the particular tetrad already belongs to this class then the distance is zero and the presentation of this tetrad with respect to the symmetry axes can be obtained. If the distance turns out to be positive, the algorithm provides a measure to see how close it is to this symmetry class. Moreover, the closest element of this class to it is also determined. This applies in cases where the tetrad is not ideal due to scattering of its measurement. The algorithm is entirely algebraic and applies to all symmetry classes, although the isotropic and the cubic class need a different treatment from all other classes.


Symmetry ◽  
2020 ◽  
Vol 12 (4) ◽  
pp. 674
Author(s):  
Houssam Abdoul-Anziz ◽  
Nicolas Auffray ◽  
Boris Desmorat

We determine the different symmetry classes of bi-dimensional flexoelectric tensors. Using the harmonic decomposition method, we show that there are six symmetry classes. We also provide the matrix representations of the flexoelectric tensor and of the complete flexoelectric law, for each symmetry class.


2012 ◽  
Vol 05 (03) ◽  
pp. 1250046
Author(s):  
Yousef Zamani ◽  
Mohammad Shahryari

The notion of Cartesian symmetry classes is introduced in [T. G. Lei, Notes on Cartesian symmetry classes and generalized trace functions, Linear Algebra Appl.292 (1999) 281–288]. In this paper, we discuss these classes and compute the dimensions of these classes in terms of the fixed point character of Sm. Also, we give a formula for the dimension of Cartesian symmetry class Vχ(G) in terms of the rank of an idempotent matrix M(χ). Some properties of generalized trace functions of a matrix are concluded.


Pramana ◽  
2021 ◽  
Vol 95 (3) ◽  
Author(s):  
Y R Kartik ◽  
Ranjith R Kumar ◽  
S Rahul ◽  
Sujit Sarkar

2006 ◽  
Vol 33 (10) ◽  
pp. 893 ◽  
Author(s):  
Hendrik Bargel ◽  
Kerstin Koch ◽  
Zdenek Cerman ◽  
Christoph Neinhuis

The cuticle is the main interface between plants and their environment. It covers the epidermis of all aerial primary parts of plant organs as a continuous extracellular matrix. This hydrophobic natural composite consists mainly of the biopolymer, cutin, and cuticular lipids collectively called waxes, with a high degree of variability in composition and structure. The cuticle and cuticular waxes exhibit a multitude of functions that enable plant life in many different terrestrial habitats and play important roles in interfacial interactions. This review highlights structure–function relationships that are the subjects of current research activities. The surface waxes often form complex crystalline microstructures that originate from self-assembly processes. The concepts and results of the analysis of model structures and the influence of template effects are critically discussed. Recent investigations of surface waxes by electron and X-ray diffraction revealed that these could be assigned to three crystal symmetry classes, while the background layer is not amorphous, but has an orthorhombic order. In addition, advantages of the characterisation of formation of model wax types on a molecular scale are presented. Epicuticular wax crystals may cause extreme water repellency and, in addition, a striking self-cleaning property. The principles of wetting and up-to-date concepts of the transfer of plant surface properties to biomimetic technical applications are reviewed. Finally, biomechanical studies have demonstrated that the cuticle is a mechanically important structure, whose properties are dynamically modified by the plant in response to internal and external stimuli. Thus, the cuticle combines many aspects attributed to smart materials.


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