  We consider a high density region of the helium phase diagram, where the
nuclei form a Bose-Einstein condensate rather than a classical plasma or a
crystal. Helium in this phase may be present in helium-core white dwarfs. We
show that in this regime there is a new gapless quasiparticle not previously
noticed, arising when the constraints imposed by gauge symmetry are taken into
account. The contribution of this quasiparticle to the specific heat of a white
dwarf core turns out to be comparable in a range of temperatures to the
contribution from the particle-hole excitations of the degenerate electrons.
The specific heat in the condensed phase is two orders of magnitude smaller
than in the uncondensed plasma phase, which is the ground state at higher
temperatures, and four orders of magnitude smaller than the specific heat that
an ion lattice would provide, if formed. Since the specific heat of the core is
an important input for setting the rate of cooling of a white dwarf star, it
may turn out that such a change in the thermal properties of the cores of
helium white dwarfs has observable implications.
