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Recently, it was established that there exists a direct relation between thenon-Hermitian skin effects, -strong dependence of spectra on boundaryconditions for non-Hermitian Hamiltonians-, and boundary zero modes forHermitian topological insulators. On the other hand, in terms of the spectraltheory, the skin effects can also be interpreted as instability of spectra fornonnormal (non-Hermitian) Hamiltonians. Applying the latter interpretation tothe former relation, we develop a theory of zero modes with quantum anomaly forgeneral Hermitian lattice systems. Our theory is applicable to a wide range ofsystems: Majorana chains, non-periodic lattices, and long-range hoppingsystems. We relate exact zero modes and quasi-zero modes of a Hermitian systemto spectra and pseudospectra of a non-Hermitian system, respectively. Thesezero and quasi-zero modes of a Hermitian system are robust against a class ofperturbations even if there is no topological protection. The robustness ismeasured by nonnormality of the corresponding non-Hermitian system. We alsopresent explicit construction of such zero modes by using a graphicalrepresentation of lattice systems. Our theory reveals the presence ofnonnormality-protected zero modes, as well as the usefulness of thenonnormality and pseudospectra as tools for topological and/or non-Hermitianphysics.

Hermitian zero modes protected by nonnormality: Application of pseudospectra