  We show the existence of Fano resonances in the context of near-field
radiative heat transfer, which enable the strong suppression and enhancement of
the spectral heat flux at specific wavelengths. This is demonstrated by making
use of the plasmon-phonon coupling in a symmetric nano cavity composed by a
polaritonic material coated with a metallic layer, in which each side of the
cavity is kept at different temperatures. The strong hybridization of the
plasmonic and phononic modes sustained by the multilayer structure is
determined by the matching of their polarization. This leads to the opening of
a thermal band gap, where heat transfer in the coupled system is inhibited for
all wave vectors and for wavelengths at which the individual constituents are
thermally transmissive.
