  White dwarf stars composed of carbon, oxygen or heavier elements are expected
to crystallize as they cool down below certain temperatures. Yet, simple
arguments suggest that the helium white dwarf cores may not solidify, mostly
because of zero-point oscillations of the helium ions that would dissolve the
crystalline structure. We argue that the interior of the helium dwarfs may
instead form a macroscopic quantum state in which the charged helium-4 nuclei
are in a Bose-Einstein condensate, while the relativistic electrons form a
neutralizing degenerate Fermi liquid. We discuss the electric charge screening,
and the spectrum of this substance, showing that the bosonic long-wavelength
fluctuations exhibit a mass gap. Hence, there is a suppression at low
temperatures of the boson contribution to the specific heat -- the latter being
dominated by the specific heat of the electrons near the Fermi surface. This
state of matter may have observational signatures.
