The non-Hermitian skin effect with three-dimensional long-range coupling

We study the non-Hermitian skin effect in a three-dimensional system of finitely many subwavelength resonators with an imaginary gauge potential. We introduce a discrete approximation of the eigenmodes and eigenfrequencies of the system in terms of the eigenvectors and eigenvalues of the so-called gauge capacitance matrix \mathcal{C}_{N}^{\gamma} , which is a dense matrix due to long-range interactions in the system. Based on translational invariance of this matrix and the decay of its off-diagonal entries, we prove the condensation of the eigenmodes at one edge of the structure by showing the exponential decay of its pseudo-eigenvectors. In particular, we consider a range k -approximation to keep the long-range interaction to a certain extent, thus obtaining a k -banded gauge capacitance matrix \mathcal{C}_{N,k}^{\gamma} . Using techniques for Toeplitz matrices and operators, we establish the exponential decay of the pseudo-eigenvectors of \mathcal{C}_{N,k}^{\gamma} and demonstrate that they approximate those of the gauge capacitance matrix \mathcal{C}_{N}^{\gamma} well. Our results are numerically verified. In particular, we show that long-range interactions affect only the first eigenmodes in the system. As a result, a tridiagonal approximation of the gauge capacitance matrix, similar to the nearest-neighbour approximation in quantum mechanics, provides a good approximation for the higher modes.