Low-temperature mass spectrum in the Ising spin glass

We study the spectrum of the Hessian of the Sherrington-Kirkpatrick model near T = 0, whose eigenvalues are the masses of the bare propagators in the expansion around the mean-field solution. In the limit T ≪ 1 two regions can be identified. The first for x close to 0, where x is the Parisi replica-symmetry-breaking scheme parameter. In this region the spectrum of the Hessian is not trivial, and maintains the structure of the full replica-symmetry-breaking state found at higher temperatures. In the second region T ≪ x ≤ 1 as T → 0, the bands typical of the full replica-symmetry-breaking state collapse and only two eigenvalues are found: a null one and a positive one. We argue that this region has a droplet-like behavior. In the limit T → 0 the width of the full replica-symmetry-breaking region shrinks to zero and only the droplet-like scenario survives.