  We show that soft spheres interacting with a linear ramp potential when
overcompressed beyond the jamming point fall in an amorphous solid phase which
is critical, mechanically marginally stable and share many features with the
jamming point itself. In the whole phase, the relevant local minima of the
potential energy landscape display an isostatic contact network of perfectly
touching spheres whose statistics is controlled by an infinite lengthscale.
Excitations around such energy minima are non-linear, system spanning, and
characterized by a set of non-trivial critical exponents. We perform numerical
simulations to measure their values and show that, while they coincide, within
numerical precision, with the critical exponents appearing at jamming, the
nature of the corresponding excitations is richer. Therefore, linear soft
spheres appear as a novel class of finite dimensional systems that
self-organize into new, critical, marginally stable, states.
