Thesis title: Lattice dynamics in systems with broken time reversal symmetry
In this thesis, the effects of time reversal symmetry breaking on lattice dynamics are investigated under both adiabatic and non-adiabatic conditions in insulators. Even under adiabatic conditions, the adiabatic force constant matrix - usually used to determine vibrational resonances - does not account for these effects, which derive from the molecular Berry curvature. The latter exerts an effective Lorentz force in the equation of motion for the nuclear system and it represents the low-frequency expansion of the frequency-dependent force constant matrix, which has to be accounted for entirely if the phonons resonate with electronic transitions (non-adiabatic case). These effects are studied in the Haldane model, a prototypical topologically non-trivial time-reversal symmetry-breaking two-dimensional system. Here, under adiabatic conditions, degenerate optical phonon modes are split into two circularly polarised modes with opposite handedness. The splitting is significantly enhanced in the non-trivial state, due to the connection between the molecular and electronic Berry curvatures at the valleys. When phonons and electronic excitations have the same energy, the lattice excitations resonate with different valleys according to their handedness in relation to the valley electronic properties, with an enhancement in the topological non-trivial state.
Moreover, we address the relation between topological states in two-dimensional systems and the infrared activity of vibrational resonances, quantified by the Born effective charges. Owing to the relation between vibrational and electronic responses in the valleys and to the parity under time-reversal symmetry, the Born effective charges display large discontinuous jumps concomitant with topological phase transitions in both the Haldane model and in the time-reversal symmetric Kane-Mele model. The latter is a prototype for the two-dimensional quantum spin Hall insulators, realised in materials like germanene and jacutingaite. In these two systems, as predicted by the model, the ab initio calculations show a jump up to 2 in the Born effective charges, manifesting in large drastic changes in the infrared spectrum associated with the transition in the topological state. The results are also robust in the presence of dynamical effects, relevant when the electronic energy gap matches the phonon frequency. In this case, the in-plane phonon shows a Fano profile in the optical conductivity with substantial differences in intensity and shape between different topological phases. The infrared response of vibrational excitations, thus, enables the discrimination between different topological states and could provide an alternative detection method for these states.
Throughout the thesis, tight-binding methods are broadly used to assess vibrational responses. The introduction of a frequency-dependent electron-phonon coupling in these models requires a careful treatment to recover all-electron responses, quantified by the fulfilment of the sum rule for the force constant matrix and the Born effective charges. We find that the non-locality of the Hamiltonian implies the presence of Peierls-like phases that depend on the nuclear velocities, previously neglected. The correction is examined in metallic doped gapped graphene, finding excellent agreement of the Born effective charges with ab initio calculations, and in the Haldane model when phonons and interband electronic transitions resonate.