  We study the phase transitions in a one dimensional Bose-Einstein condensate
on a ring whose atomic scattering length is modulated periodically along the
ring. By using a modified Bogoliubov method to treat such a nonlinear lattice
in the mean field approximation, we find that the phase transitions are of
different orders when the modulation period is 2 and greater than 2. We further
perform a full quantum mechanical treatment based on the time-evolving block
decimation algorithm which confirms the mean field results and reveals
interesting quantum behavior of the system. Our studies yield important
knowledge of competing mechanisms behind the phase transitions and the quantum
nature of this system.
