The behavior of higher-order boundary states in non-Hermitian systems is elusive, thus finding the mechanism behind these states is both essential and significant. Here, we uncover a novel mechanism that induces anomalous higher-order boundary states. The mechanism originates from the sensitivity of the non-normal boundary Hamiltonian to intrinsic perturbations, which refer here to the influence of the bulk on the topological boundaries. Based on the mechanism, we reveal a new kind of phase transition, i.e., the transition between hybrid skin-topological states and scale-free topological boundary states. We also find that scale-free topological boundary states exhibit size-dependent spectra, influencing the existence of higher-order topological boundary states. Unlike conventional hybrid skin-topological states or the higher-order non-Hermitian skin effect, the above two kinds of anomalous higher-order boundary states exhibit size-dependent characteristics. Our work opens a new horizon for the control of higher-order boundary states and topological properties of non-Hermitian systems.

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• Edge states
• Topological phases of matter
• Non-Hermitian systems
• Tight-binding model