Terahertz dynamics of ferroelectric vortices from first principles

A first-principles-based effective Hamiltonian is used to reveal dynamics of vortices in ferroelectrics. In addition to the "usual" dielectric modes that are generated by the fluctuation of the electrical polarization, additional toroidic modes, resulting from the electric toroidal moment fluctuations, are also found in the THz regime. Such latter modes can have their own dynamics, with a resonant frequency that softens via a square-root law when the temperature approaches the critical temperature at which electric vortices form. These dynamics are also found to be governed by the fluctuation of the self-organized azimuthal component of individual electric dipoles. Toroidic modes thus behave as pendulums, in contrast to springs that represent polarization dynamics.