Symmetry (geometry)

About: Symmetry (geometry) is a research topic. Over the lifetime, 7710 publications have been published within this topic receiving 113087 citations.

Topological nodal line semimetals

Abstract: We review the recent, mainly theoretical, progress in the study of topological nodal line semimetals in three dimensions. In these semimetals, the conduction and the valence bands cross each other along a one-dimensional curve in the three-dimensional Brillouin zone, and any perturbation that preserves a certain symmetry group (generated by either spatial symmetries or time-reversal symmetry) cannot remove this crossing line and open a full direct gap between the two bands. The nodal line(s) is hence topologically protected by the symmetry group, and can be associated with a topological invariant. In this review, (i) we enumerate the symmetry groups that may protect a topological nodal line; (ii) we write down the explicit form of the topological invariant for each of these symmetry groups in terms of the wave functions on the Fermi surface, establishing a topological classification; (iii) for certain classes, we review the proposals for the realization of these semimetals in real materials; (iv) we discuss different scenarios that when the protecting symmetry is broken, how a topological nodal line semimetal becomes Weyl semimetals, Dirac semimetals, and other topological phases; and (v) we discuss the possible physical effects accessible to experimental probes in these materials.

Multi-Weyl topological semimetals stabilized by point group symmetry.

Abstract: We perform a complete classification of two-band k . p theories at band crossing points in 3D semimetals with n-fold rotation symmetry and broken time-reversal symmetry. Using this classification, we show the existence of new 3D topological semimetals characterized by C-4,C-6-protected double-Weyl nodes with quadratic in-plane (along k(x,y)) dispersion or C-6-protected triple-Weyl nodes with cubic in-plane dispersion. We apply this theory to the 3D ferromagnet HgCr2Se4 and confirm it is a double-Weyl metal protected by C-4 symmetry. Furthermore, if the direction of the ferromagnetism is shifted away from the [001] axis to the [111] axis, the double-Weyl node splits into four single Weyl nodes, as dictated by the point group S-6 of that phase. Finally, we discuss experimentally relevant effects including the splitting of multi-Weyl nodes by applying a C-n breaking strain and the surface Fermi arcs in these new semimetals.

A remarkable periodic solution of the three-body problem in the case of equal masses

Abstract: Using a variational method, we exhibit a surprisingly simple periodic orbit for the newtonian problem of three equal masses in the plane. The orbit has zero angular momentum and a very rich symmetry pattern. Its most surprising feature is that the three bodies chase each other around a flxed eight-shaped curve. Setting aside collinear motions, the only other known motion along a flxed curve in the inertial plane is the \Lagrange relative equilibrium" in which the three bodies form a rigid equilateral triangle which rotates at constant angular velocity within its circumscribing circle. Our orbit visits in turns every \Euler conflguration" in which one of the bodies sits at the midpoint of the segment deflned by the other two (Figure 1). Numerical computations

Facial symmetry and the perception of Beauty

Abstract: Evolutionary, as well as cultural, pressures may contribute to our perceptions of facial attractiveness. Biologists predict that facial symmetry should be attractive, because it may signal mate quality. We tested the prediction that facial symmetry is attractive by manipulating the symmetry of individual faces and observing the effect on attractiveness, and by examining whether natural variations in symmetry (between faces) correlated with perceived attractiveness. Attractiveness increased when we increased symmetry, and decreased when we reduced symmetry, in individual faces (Experiment 1), and natural variations in symmetry correlated significantly with attractiveness (Experiments 1 and 1A). Perfectly symmetric versions, made by blending the normal and mirror images of each face, were preferred to less symmetric versions of the same faces (even when those versions were also blends) (Experiments 1 and 2). Similar results were found when subjects judged the faces on appeal as a potential life partner, suggesting that facial symmetry may affect human mate choice. We conclude that facial symmetry is attractive and discuss the possibility that this preference for symmetry may be biologically based.