The skin effect, which is unique to non-Hermitian systems, can generate an extensive number of eigenstates localized near the boundary in an open geometry. Here, we propose that in two dimensions (2D) and three dimensions (3D) other quantities besides charge density are susceptible to the skin effect. We show that 2D and 3D models that are a hybrid between topological insulators and skin-effect systems can have a topological skin effect where an extensive number of topological modes, and the corresponding bulk topological invariant, are pinned to the surface. A key example, which we call the quantum skin Hall effect is constructed from layers of Chern insulators and exhibits an extensive Hall conductance and number of chiral modes bound to surfaces normal to the stack of layers. The same procedure is further extended to other symmetry classes to illustrate that a variety of one-dimensional and 2D topological invariants ($\mathbb{Z}$ or ${\mathbb{Z}}_2)$ are subject to the skin effect. Indeed, we also propose a hybrid 2D system that exhibits an extensive number of topological corner modes and may be more easily realized in metamaterial experiments.

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• Chern insulators
• Topological insulators
• Non-Hermitian systems