Fig. 3. Non-Hermitian skin effect via the odd micropolar elasticity.
a The vibrational spectrum for the flexural mode of a metabeam with periodic boundary conditions and odd micropolar modulus P = 3Π. The black line results from the continuum theory given by Eq. (28). The data points are obtained via fully piezoelectrically coupled simulations in COMSOL with the hue indicating the wavenumber kL, where L is the unit cell length. For the full spectrum plotted as a function of k in the continuum theory and in the numerics, see Fig. 7 and S2, respectively. The inset compares the continuum theory and simulations for small wavenumbers . b The inverse penetration depth κ for real ω in a medium with open boundary conditions. The points are the results of COMSOL simulations, the black lines are Eq. (29), and the dark lines are the result of the transfer matrix method, see Supplementary Note 3. c The localized states are connected to a topological index ν(ω). The periodic boundary spectrum for P > 0, P = 0, and P < 0 are represented schematically by the solid lines. The arrows indicate the direction of increasing k. For a given frequency ω, the winding number ν(ω) of the periodic boundary spectrum indicates the presence of a localized mode. d The localization of eigenmode at the value of ω denoted by the star in (c) is schematically illustrated.