Solid State Theory pp 281-292 | Cite as

# Appendices

## Abstract

Geometrical operations (translation, rotation, inversion) which leave a geometrical object (here the crystal lattice) invariant, are symmetry operations. Mathematically they form a group, the symmetry group of the crystal: for the translations it is the *translation group*, for the rotations, inversion, and their combinations it is the *point group*. The number *g* of elements in the group is its order. The symmetry of a system implies the invariance of the system hamiltonian *H* (be it for phonons, electrons, or magnons) under unitary operations corresponding to the geometrical operations of the symmetry group. These unitary operations form a group which is isomorphic to the symmetry group. When applied to a set of eigenfunctions of *H* this set is transformed into another set of eigenfunctions, which can be represented as a linear combination of the former ones. The eigenfunctions of a degenerate eigenvalue transform among each other and form an invariant subspace in the Hilbert space of *H*. In a chosen basis, these unitary operations can be formulated as matrices which define another group isomorphic to the symmetry group. For a proper choice of the basis, all matrices of the matrix representation have block-diagonal form with the dimension of the block matrices indicating the degeneracy of the invariant subspaces.

## Keywords

Symmetry Group Irreducible Representation Invariant Subspace Point Group Symmetry Operation## Preview

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