Two-phonon pseudogap in the Klein-Gordon lattice

– The energy spectrum of the quantum Klein-Gordon (KG) lattice is computed numerically for model parameters relevant to optical phonon spectra. A pairing of phonon states is found when nonlinearity is significant, which agrees with other studies on different quantum lattice models [1, 2]. It results from the lattice anharmonicity, the magnitude of which is quantified by the binding energy of phonon bound states. Our work focuses on the case of weak anharmonicity, i.e., the phonon binding energy is weaker than the single phonon band width. We find that the phonon pairs dissociate at the center of the lattice Brillouin zone, whereas at the edge the binding energy remains comparable to the width of the single phonon band. Consequently, a weak nonlinearity is characterized by a pseudogap in the energy spectrum of two-phonon states. The phonon frequencies are some fundamental quantities that provide information about atomic interactions and local structure in crystals as well as in polymers or proteins. At the atomic scale, the phonon frequencies are formally derived from the quadratic expansion of the potential energy with respect to atomic displacements. Actually a more accurate description of the potential energy might be obtained by a higher order expansion which involves some non-quadratic terms. Those terms are usually referred to as nonlinear terms because they yield some forces that are not proportional to displacements. In the early seventies, the nonquadratic contribution to the energy has been quantified in molecular crystals such as CO2, N2O and OCS (see Ref. [3] and Refs. therein). The nonlinearity was then identified by some anharmonic peaks in the infrared spectrum. In solid H2, similar infrared resonances were earlier interpreted [4] as a signature of phonons pairing. Currently the list of materials in which phonon bound states occur is still growing [5, 6]. In addition, some controversial cases, as for instance acetanilide [7] or the spectral branch ∆′2(LO) of diamond [8, 9] can be mentioned as still open questions. In the present study, we point out a possible difficulty for the vibration spectrum interpretation which could occur when nonlinearity is not sufficiently strong to separate the anharmonic resonances from the harmonic energy regions. Then, the hybridization between bound and unbound phonons is shown to imply a pseudogap in the lattice energy spectrum. The anharmonicity of molecular crystals has been theoretically studied by V.M. Agranovich [1], who proposed a qualitative approach where some boson quasi-particles model the