Negative refraction and rotons in the relativistic Bose gas

We investigate the dispersion of a classical electromagnetic field in a relativistic ideal gas of charged bosons using scalar quantum electrodynamics at finite temperature and charge density. We derive the effective electromagnetic responses and the electromagnetic propagation modes that characterize the gas as a left-handed material with negative effective index of refraction $n_{\rm eff}=-1$ below the transverse plasmon frequency. In the condensed phase, we show that the longitudinal plasmon dispersion relation exhibits a roton-type local minimum that disappears at the transition temperature.