Topological directed amplification

A phenomenon of topological directed amplification of certain initial perturbations is revealed theoretically to emerge in a class of asymptotically stable skin-effect lattices described by nonnormal Toeplitz operators $H_g$ with positive ``numerical ordinate" $\omega(H_g)>0$. Nonnormal temporal evolution, even in the presence of global dissipation, is shown to manifest a counterintuitive transient phase of edge-state amplification -- a behavior, drastically different from the asymptote, that spectral analysis of $H_g$ fails to directly reveal. A consistent description of the effect is provided by the general tool of ``pseudospectrum", and a quantitative estimation of the maximum power amplification is provided by the {\it Kreiss constant}. A recipe to determine an optimal initial condition that will attain maximum amplification power is given by singular value decomposition of the propagator $e^{-i H_g t}$. It is further predicted that the interplay between nonnormality and nonlinearity in a skin-effect laser array can facilitate narrow-emission spectra with scalable stable-output power.