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

TL;DR

This study investigates the effects of small spin splitting on the $C6B2B2B4$ modular symmetry in quantum Hall systems, revealing deviations in critical conductivities and universal scaling behaviors despite broken symmetry.

Contribution

It demonstrates how small spin splitting reduces the modular symmetry in quantum Hall systems and affects the scaling laws and critical conductivities.

Findings

Distorted flowlines and shifted critical points observed.

Universal scaling persists despite symmetry reduction.

Hall conductivity follows the scaling law even when other laws fail.

Abstract

Magnetic-field-induced phase transitions were studied with a two-dimensional electron AlGaAs/GaAs system. The temperature-driven flow diagram shows the features of the $\Gamma$(2) modular symmetry, which includes distorted flowlines and shiftted critical point. The deviation of the critical conductivities is attributed to a small but resolved spin splitting, which reduces the symmetry in Landau quantization. [B. P. Dolan, Phys. Rev. B 62, 10278.] Universal scaling is found under the reduction of the modular symmetry. It is also shown that the Hall conductivity could still be governed by the scaling law when the semicircle law and the scaling on the longitudinal conductivity are invalid. *corresponding author:yhchang@phys.ntu.edu.tw