  In the half-filled zero-energy Landau level of bilayer graphene, competing
phases with spontaneously broken symmetries and an intriguing quantum critical
behavior have been predicted. Here we investigate signatures of these
broken-symmetry phases in thermal transport measurements. To this end we
calculate the spectrum of spin and valley waves in the $\nu=0$ quantum Hall
state of bilayer graphene. The presence of Goldstone modes enables heat
transport even at low temperatures, which can serve as compelling evidence for
spontaneous symmetry breaking. By varying external electric and magnetic fields
it is possible to determine the nature of the symmetry breaking and
temperature-dependent measurements may yield additional information about
gapped modes.
