Fig. 1. Four different types of critical transitions.

Hermitian phase transitions (a) are marked by gap closures along the real line. In non-Hermitian cases (2nd to 4th rows, axis labels omitted), spectral phase transitions can take more sophisticated possibilities in the 2D complex energy plane. For instance, the spectral topology can change under line gap closures (b) or shrink continuously to a point and re-emerge in a different topological configuration (c), without the gap ever closing³⁸. The spectrum continuously passes through a gapless or point-like regime in the first three cases, as indicated by the black arrows. The critical non-Hermitian skin effect (d), however, is special in that OBC spectrum in the thermodynamic limit, denoted E_∞, jumps discontinuously from one configuration (left), to a different configuration (middle), and to another (right) as certain parameter changes from  −ϵ to 0 (critical border), and to ϵ, for an arbitrarily small ϵ, without ever interpolating between the configurations even though the parameter is continuously tuned.