Roberge-Weiss phase transitions and extended Z3 symmetry
  Using the Polyakov extended Nambu-Jona-Lasinio (PNJL) model with imaginary chemical potential, the relation between the Roberge-Weiss (RW) phase transition and the extended Z3 symmetry is studied. At low temperature, there is approximate continuous symmetry under the phase transformation of the Polyakov loop with the shift of the imaginary chemical potential. Due to this continuous symmetry, the Polyakov loop can oscillate smoothly as the imaginary chemical potential increases. At high temperature, this continuous symmetry is broken to an exact discrete symmetry, the extended Z3 symmetry, and the Polyakov loop can not oscillate smoothly. This symmetry breaking of the continuous symmetry causes a discontinuity of the Polyakov loop. That is the RW phase transition. 
H. Kouno, Y. Sakai, K. Kashiwa, M. Matsuzaki and M. Yahiro