Mesoscopic charge density wave in a magnetic flux

The stability of a Charge Density Wave (CDW) in a one-dimensional ring pierced by a Aharonov-Bohm flux is studied in a mean-field picture. It is found that the stability depends on the parity of the number N of electrons. When the size of the ring becomes as small as the coherence length xi, the CDW gap increases for even N and decreases for odd N. Then when N is even, the CDW gap decreases with flux but it increases when N is odd. The variation of the BCS ratio with size and flux is also calculated. We derive the harmonics expansion of the persistent current in a presence of a finite gap.