<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="script">PT</mml:mi></mml:math>-symmetric phase transition and unidirectional accumulation of eigenstates in a non-Hermitian system with a single impurity

Considering the extreme sensitivity of a non-Hermitian system to boundary conditions, we introduce a single impurity as an effective boundary and investigate its impact on the system. In the absence of impurities, we observe the non-Hermitian skin effect and topologically nontrivial phase, characterized by the spectral and non-Bloch winding numbers, respectively. After introducing a single impurity, we investigate multiple transitions between $\mathcal{PT}$-symmetry-broken and -unbroken phases from both numerical and analytical perspectives. Moreover, we reveal that the introduction of a single impurity disrupts the translational invariance of the system, resulting in the unidirectional accumulation of bulk states around the impurity and the formation of bound states with purely imaginary eigenvalues localized at the impurity. Meanwhile, the directions of unidirectional accumulation of bulk states and the ranges of existence of bound states are both regulated by the intracell hopping strength and on-site potential strength. Our work demonstrates the sensitivity of non-Hermitian systems to a single impurity and promotes the investigation of non-Hermitian systems with impurities.