  Using a continuum Dirac theory, we study the density and spin response of
zigzag edge terminated graphene ribbons subjected to edge potentials and Zeeman
fields. Our analytical calculations of the density and spin responses of the
closed system (fixed particle number) to the static edge fields, show a highly
nonlinear Weber-Fechner type behavior where the response depends
logarithmically on the edge potential. The dependence of the response on the
size of the system (e.g. width of a nanoribbon) is also uncovered. Zigzag edge
graphene nanoribbons, therefore, provide a realization of response of organs
such as the eye and ear that obey Weber-Fechner law. We validate our analytical
results with tight binding calculations. These results are crucial in
understanding important effects of electron-electron interactions in graphene
nanoribbons such as edge magnetism etc., and also suggest possibilities for
device applications of graphene nanoribbons.
