A Hamiltonian approach for obtaining irreducible projective representations and the k ⋅ p perturbation for anti-unitary symmetry groups. (7th June 2021) - Yang, Zhen-Yuan; Yang, Jian; Fang, Chen; Liu, Zheng-Xin
Abstract: As is known, the irreducible projective representations (Reps) of anti-unitary groups contain three different situations, namely, the real, the complex and quaternionic types with torsion number 1, 2, 4 respectively. This subtlety increases the complexity in obtaining irreducible projective Reps of anti-unitary groups. In the present work, a physical approach is introduced to derive the condition of irreducibility for projective Reps of anti-unitary groups. Then a practical procedure is provided to reduce an arbitrary projective Rep into direct sum of irreducible ones. The central idea is to construct a Hermitian Hamiltonian matrix which commutes with the representation of every group element g ∈ G, such that each of its eigenspaces forms an irreducible representation space of the group G . Thus the Rep is completely reduced in the eigenspaces of the Hamiltonian. This approach is applied in the k ⋅ p effective theory at the high symmetry points (HSPs) of the Brillouin zone for quasi-particle excitations in magnetic materials. After giving the criterion to judge the power of single-particle dispersion around an HSP, we then provide a systematic procedure to construct the k ⋅ p effective model.