Low Temperature Specific Heat of some Quantum Mean Field glassy phases

We investigate analytically the low temperature behavior of the specific heat $C_v(T)$ for a large class of quantum disordered models within Mean Field approximation. This includes the vibrational modes of a lattice pinned by impurity disorder in the quantum regime, the quantum spherical $p$-spin-glass and a quantum Heisenberg spin glass. We exhibit a general mechanism, common to all these models, arising from the so-called marginality condition, responsible for the cancellation of the linear and quadratic contributions in $T$ in the specific heat. We thus find for all these models the Mean Field result $C_v(T) \propto T^3$.