  We investigate the properties of the glass phase of a recently introduced
spin glass model of soft spins subjected to an anharmonic quartic local
potential, which serves as a model of low temperature molecular or soft
glasses. We solve the model using mean field theory and show that, at low
temperatures, it is described by full replica symmetry breaking (fullRSB). As a
consequence, at zero temperature the glass phase is marginally stable. We show
that in this case, marginal stability comes from a combination of both soft
linear excitations -- appearing in a gapless spectrum of the Hessian of linear
excitations -- and pseudogapped non-linear excitations -- corresponding to
nearly degenerate two level systems. Therefore, this model is a natural
candidate to describe what happens in soft glasses, where quasi localized soft
modes in the density of states appear together with non-linear modes triggering
avalanches and conjectured to be essential to describe the universal
low-temperature anomalies of glasses.
