Abstract
Nonreciprocal nonhermitian systems provide an unconventional localization mechanism of topological zero modes via the nonhermitian skin effect. While fundamental theoretical characterizations of this effect involve the biorthogonal system of right and left eigenmodes, the recent demonstration of this effect for a zero mode in a robotic metamaterial (Ghatak et al., arXiv:1907.11619) is based on the direct experimental observation of the conventional right eigenvectors. Here I show that such nonreciprocal mechanical metamaterials reveal their underlying biorthogonality in the directly observable response of the system to external excitation. Applied to the ground-breaking experiment, this nonreciprocal response theory predicts that the zero-mode skin effect goes along an extended phase where the system is highly sensitive to physical perturbations, leading to a diverging response in the limit of a large system.
Highlights
Nonreciprocal non-Hermitian mechanical metamaterials are a recent innovation [1,2] that enable the experimental study of phenomena arising from the interplay of two principal notions of broken time-reversal symmetry in conservative and dissipative systems
These signatures are revealed when one develops the response theory for nonreciprocal media subjected to physical external excitation, which I here exemplify for a general class of systems compassing the robotic metamaterials
Probing the dynamical response of nonHermitian nonreciprocal metamaterials gives directly observable insights into the right and left eigenmodes, while the overall sensitivity of the system is governed by the full biorthogonal interplay between both sets of modes
Summary
Nonreciprocal non-Hermitian mechanical metamaterials are a recent innovation [1,2] that enable the experimental study of phenomena arising from the interplay of two principal notions of broken time-reversal symmetry in conservative and dissipative systems. As I point out in this paper, the left eigenmodes as well as the complete biorthogonal interplay both leave clear signatures that can be directly observed in experiments These signatures are revealed when one develops the response theory for nonreciprocal media subjected to physical external excitation, which I here exemplify for a general class of systems compassing the robotic metamaterials. The non-Hermitian skin effect of the zero mode becomes linked with a phase transition, where the sensitivity of the system to low-frequency excitations diverges in the limit of a large system This extreme sensitivity, which occurs across the whole skin-effect phase and is independent of any spectral singularities, is described by the formal analog of the Petermann factor from quantum-limited noise theory [12,13,14,15] and applies generally to a wide class of nonreciprocal non-Hermitian media.
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