Higher-order non-Hermitian skin effect

The non-Hermitian skin effect is a unique feature of non-Hermitian systems, in which an extensive number of boundary modes appear under the open boundary conditions. Here, we discover higher-order counterparts of the non-Hermitian skin effect that exhibit new boundary physics. In two-dimensional systems with the system size $L \times L$, while the conventional (first-order) skin effect accompanies $O\,( L^{2} )$ skin modes, the second-order skin effect accompanies $O\,( L )$ corner skin modes. This also contrasts with Hermitian second-order topological insulators, in which only $O\,( 1 )$ corner zero modes appear. Moreover, for the third-order skin effect in three dimensions, $O\,( L )$ corner skin modes appear from all $O\,( L^{3} )$ modes. We demonstrate that the higher-order skin effect originates from intrinsic non-Hermitian topology protected by spatial symmetry. We also show that it accompanies the modification of the non-Bloch band theory in higher dimensions.