Fulde-Ferrell-Larkin-Ovchinnikov states in a superconducting ring with   magnetic fields: Phase diagram and the first-order phase transitions
  We find the angular Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) states (or the twisted kink crystals) in which a phase and an amplitude of a pair potential modulate simultaneously in a quasi-onedimensional superconducting ring with a static Zeeman magnetic field applied on the ring and static Aharonov-Bohm magnetic flux penetrating the ring. The superconducting ring with magnetic flux produces a persistent current, whereas the Zeeman split of Fermi energy results in the spatial modulation of the pair potential. We show that these two magnetic fields stabilize the FFLO phase in a large parameter region of the magnetic fields. We further draw the phase diagram with the two kinds of first-order phase transitions; one corresponds to phase slips separating the Aharonov-Bohm magnetic flux, and the other separates the number of peaks of the pair amplitude for the Zeeman magnetic field. 
Ryosuke Yoshii, Satoshi Takada, Shunji Tsuchiya, Giacomo Marmorini,   Hisao Hayakawa, Muneto Nitta
cond-mat.supr-con hep-th quant-ph