Three-band non-Hermitian non-Abelian topological insulator with PT symmetry

Non-Abelian topological charges have recently been proposed to characterize the noncommutative topological properties in multiple-band-gap systems. In this work, we investigate a non-Abelian system based on a three-band tight-binding model characterized by quaternions. By introducing nonreciprocal hopping, the system becomes non-Hermitian yet still preserves $\mathcal{PT}$ symmetry within a range of hopping parameters. As the hopping strength increases, the $\mathcal{PT}$ symmetry is spontaneously broken. We find that the critical hopping values of $\mathcal{PT}$ transitions for quaternion charges ${\ifmmode\pm\else\textpm\fi{}i,\ifmmode\pm\else\textpm\fi{}k,\ensuremath{-}1}$ are different from that of the $\ifmmode\pm\else\textpm\fi{}j$ case. Non-Hermitian quaternion topological charges are analytically analyzed using the Berry-Wilczek-Zee (BWZ) phase. By diagonalizing the BWZ phase matrix, we discover that its eigenvalues undergo a real-to-imaginary $\mathcal{PT}$ transition. This corresponds to the real-to-imaginary transition of the associated eigenvalues of the Hamiltonian. However, the boundary states in the finite system remain robust against random disorder only when the $\mathcal{PT}$ symmetry is preserved. Furthermore, we examine the existence of domain wall states between two subsystems carrying different topological charges to verify the non-Abelian quotient relation in the non-Hermitian situation. It is revealed that domain wall states are no longer robust against random disorder once the $\mathcal{PT}$ symmetry is broken. Our work deepens the physical understanding of the non-Abelian topology in the non-Hermitian system with $\mathcal{PT}$ symmetry.