Abstract
AbsractSymmetry degree is utilized to characterize the asymmetry of a physical system with respect to a symmetry group. The scalar form of symmetry degree (SSD) based on Frobenius-norm has been introduced recently to present a quantitative description of symmetry. Here we present the vector form of the symmetry degree (VSD) which possesses more advantages than the SSD. Mathematically, the dimension of VSD is defined as the conjugacy class number of the symmetry group, the square length of the VSD gives rise to the SSD and the direction of VSD is determined by the orders of the conjugacy classes. The merits of applying VSD both for finite and infinite symmetry groups include the additional information of broken symmetry operators with single symmetry breaking perturbation, and the capability of distinguishing distinct symmetry breaking perturbations which exactly give rise to degenerate SSD. Additionally, the VSD for physical systems under symmetry breaking perturbations can be regarded as a projection of the initial VSD without any symmetry breaking perturbations, which can be described by an evolution equation. There are the same advantages by applying VSD for the accidental degeneracy and spontaneous symmetry breaking.
Highlights
Symmetry degree is utilized to characterize the asymmetry of a physical system with respect to a symmetry group
To avoid this artificial missing, we propose the vector form of the symmetry degree (VSD) instead, which is obtained by dividing the scalar form of symmetry degree (SSD) according to the conjugacy classes of the symmetry group
One more important property for VSD is that the information of symmetry degree on individual transformations in the same conjuagcy class is stored into the component of the VSD, which is illustrated by the direction of the VSD
Summary
Symmetry degree is utilized to characterize the asymmetry of a physical system with respect to a symmetry group. In most of cases the stark effect resulting from the electric field is sufficiently weak, D3h symmetry is still applied to obtain the energy spectrum and the electronic dynamics in such system In this sense, the symmetry needs a continuous quantitative description for physical systems. More information of symmetry degree on an individual transformation is lost during this averaging calculation, which is supposed to reveal the delicate nature of symmetry To avoid this artificial missing, we propose the vector form of the symmetry degree (VSD) instead, which is obtained by dividing the SSD according to the conjugacy classes of the symmetry group. The accurate definition of the continuous quantitative description of the symmetry may shed light on symmetry related applications in various physics fields
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