Microscopic manifestation of the spin phase transition at filling factor 2/3

First-order phase transitions are known to be accompanied by hysteresis due to the formation of domains with different order parameters. As the transition is approached, the domain sizes increase to macroscopic dimensions. It is therefore natural to expect that hysteresis will be absent from samples of microscopic size. Here, we explore from a microscopic standpoint the hysteretic behaviour across the spin phase transition that occurs at filling 2/3 in the fractional-quantum-Hall regime. Using a single-electron transistor, we follow the evolution of localized states across the spin transition by measuring the local compressibility. Localized-state spectra clearly reveal the hysteretic behaviour accompanying the transition. Using electrostatic gating we continuously vary the size of the sample undergoing the phase transition. For submicrometre dimensions the hysteresis disappears, indicating domain sizes in excess of 500 nm.