  Structural glasses display at low temperature a set of anomalies in
thermodynamic observables. The prominent example is the linear-in-temperature
scaling of the specific heat, at odds with the Debye cubic scaling found in
crystals, due to acoustic phonons. Such an excess of specific heat in amorphous
solids is thought of arising from phenomenological soft excitations dubbed
tunneling two-level systems (TTLS). Their nature as well as their statistical
properties remain elusive from a first-principle viewpoint. In this work we
investigate the canonically quantized version of the KHGPS model, a mean-field
glass model of coupled anharmonic oscillators, across its phase diagram, with
an emphasis on the specific heat. The thermodynamics is solved in a
semiclassical expansion. We show that in the replica-symmetric region of the
model, up to the marginal glass transition line where replica symmetry gets
continuously broken, a disordered version of the Debye approximation holds: the
specific heat is dominated by harmonic vibrational excitations inducing a
power-law scaling at the transition, ruled by random matrix theory. This
mechanism generalizes a previous semiclassical argument in the literature. We
then study the marginal glass phase where the semiclassical expansion becomes
non-perturbative due to the emergence of instantons that overcome disordered
Debye behavior. Inside the glass phase, a variational solution to the instanton
approach provides the prevailing excitations as TTLS, which generate a linear
specific heat. This phase thus hosts a mix of TTLS and harmonic excitations
generated by interactions. We finally suggest to go beyond the variational
approximation through an analogy with the spin-boson model.
