An experimental study on Γ(2) modular symmetry in the quantum Hall system with a small spin splitting

Magnetic-field-induced phase transitions were studied with a two-dimensional electron AlGaAs/GaAs system. The temperature-driven flow diagram shows features of the Γ(2) modular symmetry, which includes distorted flowlines and a shifted critical point. The deviation of the critical conductivities is attributed to a small but resolved spin splitting, which reduces the symmetry in Landau quantization (Dolan 2000 Phys. Rev. B 62 10278). Universal scaling is found under the reduction of the modular symmetry. It is also shown that the Hall conductivity can still be governed by the scaling law when the semicircle law and the scaling on the longitudinal conductivity are invalid.