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.