A Hamiltonian approach for obtaining irreducible projective representations and the k ⋅ p perturbation for anti-unitary symmetry groups

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