An experimental study on $\Gamma $(2) modular symmetry in the quantum Hall system with a small spin-splitting

Magnetic-field-induced phase transitions are investigated by performing transport measurements on the two-dimensional (2D) electron system in an AlGaAs/GaAs heterostructure at low temperatures. [1] Both the semicircle law and universal two-parameter scaling are observed as the spin-splitting becomes resolved with increasing the magnetic field $B$ perpendicular to such a 2D system. The critical resistivities, however, are not of the expected universal values, and the modular symmetry is reduced from $\Gamma _{0}$(2) to $\Gamma $(2). On the other hand, $\Gamma _{0}$(2) symmetry survives at lower $B$ where the spin-splitting is unresolved. Therefore, the reduction of the modular symmetry can be due to the resolved small spin-splitting as predicted by Dolan [2]. Such a small splitting not only reduces the modular symmetry, but also breaks the scaling on the longitudinal conductivity \textit{$\sigma $}$_{xx}$ at higher temperatures. It is found that the scaling on the Hall conductivity \textit{$\sigma $}$_{xy}$ is more robust than that on \textit{$\sigma $}$_{xx}$ under a small spin gap. References [1] C. F. Huang, Y. H. Chang, H. H. Cheng, Z. P. Yang, H. D. Yeh, C. H. Hsu, C.-T. Liang, H. H. Lin, D. R. Hang, and H. H. Lin, J. Phys.:Condens. Matt. 19, 026205 (2007). [2] B. P. Dolan, Phys. Rev. B 62, 10278 (2000).