Non-Hermitian skin effects in two- and three-dimensional intertwined tight-binding lattices

With the generic non-Hermitian Hatano-Nelson Hamiltonians in real space, we investigate both analytically and numerically a multitude of non-Hermitian skin effects in a two- or three-dimensional intertwined lattice. Exact skin-mode solutions in such high-dimensional lattices as well as their topological explanations are provided, when subjected to hybrid boundary conditions or certain types of fully open boundary conditions. Under fully open boundary conditions, we particularly demonstrate several unusual non-Hermitian skin modes in either a two- or three-dimensional lattice, including the edge skin modes that can condense simultaneously on two or more edges for a two-dimensional lattice and the corner skin modes that have a much higher state density than others for a three-dimensional lattice, neither of which are achievable under hybrid boundary conditions. We anticipate that these results may inspire and facilitate the experimental studies of high-dimensional non-Hermitian physics, given their potential to realizing non-Hermitian topological lasing.