Structural glasses display at low temperature a set of anomalies in thermodynamic observables. A 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.

40 pages, 16 figures