We show that a non-Hermitian zero mode can exhibit an unusual behavior at the transition between extended and localized regimes: it displays a linearly decreasing amplitude as a function of space in a weakly coupled non-Hermitian reservoir. Through the discussion of a linear homogeneous recurrence relation, we attribute this phenomenon to the underlying non-Hermitian particle-hole symmetry and the zeroness of its energy eigenvalue. We also show that linear localization bears a strong resemblance to critical damping, even though the latter does not display linear temporal dynamics.
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