Real-space imaging of fractional quantum Hall liquids

Electrons in semiconductors usually behave like a gas-as independent particles. However, when confined to two dimensions under a perpendicular magnetic field at low temperatures, they condense into an incompressible quantum liquid. This phenomenon, known as the fractional quantum Hall (FQH) effect(1), is a quantum-mechanical manifestation of the macroscopic behaviour of correlated electrons that arises when the Landau-level filling factor is a rational fraction. However, the diverse microscopic interactions responsible for its emergence(1,2) have been hidden by its universality and macroscopic nature(3-9). Here, we report real-space imaging of FQH liquids, achieved with polarization-sensitive scanning optical microscopy using trions (charged excitons)(10-15) as a local probe for electron spin polarization. When the FQH ground state is spin-polarized, the triplet/singlet intensity map exhibits a spatial pattern that mirrors the intrinsic disorder potential, which is interpreted as a mapping of compressible and incompressible electron liquids. In contrast, when FQH ground states with different spin polarization coexist, domain structures with spontaneous quasi-long-range order emerge, which can be reproduced remarkably well from the disorder patterns using a two-dimensional random-field Ising model(16). Our results constitute the first reported real-space observation of quantum liquids in a class of broken symmetry state known as the quantum Hall ferromagnet(17-25).