  We consider the energy of the filled quasiparticle's Fermi sea of a
macroscopic superconducting ring threaded by an hc/2e-vortex, when the material
of the ring is of an unconventional pairing symmetry. The energy relative to
the one for the hc/e-vortex configuration is finite, positive, and inversely
proportional to ring's inner radius. We argue that the existence of this energy
in unconventional superconductors removes the commonly assumed degeneracy
between the odd and the even vortices, with the loss of the concomitant hc/2e
periodicity in external magnetic field as a consequence. This macroscopic
quantum effect should be observable in nanosized unconventional superconductors
with a small phase stiffness, such as deeply underdoped YBCO with Tc < 5K.
