Effects of Zeeman spin splitting on the modular symmetry in the quantum   Hall effect
  Magnetic-field-induced phase transitions in the integer quantum Hall effect are studied under the formation of paired Landau bands arising from Zeeman spin splitting. By investigating features of modular symmetry, we showed that modifications to the particle-hole transformation should be considered under the coupling between the paired Landau bands. Our study indicates that such a transformation should be modified either when the Zeeman gap is much smaller than the cyclotron gap, or when these two gaps are comparable. 
D. R. Hang, R. B. Dunford, Gil-Ho Kim, H. D. Yeh, C. F. Huang, D. A.   Ritchie, I. Farrer, Y. W. Zhang, C.-T. Liang, and Y. ...