Algebraic Impurity States in Higher Dimensional Non-Hermitian Systems

We investigate the eigenstates in the presence of a single impurity in the continuum spectrum of two-dimensional non-Hermitian systems. For generic impurity strengths, there is one special mode the amplititude of which decays algebraically from the impurity site as 1/|r|. The energy for this mode lies in the continuum of the spectrum under the periodic boundary condition. The threshold of impurity strength for the existence of these algebraic impurity states is given in terms of the energy and the gradients of energy with respect to k_x and k_y. Our work unveils the existence of impurity-induced algebraically localized modes in the continuum spectrum. This is a unique feature of higher-dimensional non-Hermitian systems since such modes are typically forbidden in Hermitian systems or non-Hermitian systems in 1D.