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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) effect1, 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 emergence1,2 have been hidden by its universality and macroscopic nature3,4,5,6,7,8,9. Here, we report real-space imaging of FQH liquids, achieved with polarization-sensitive scanning optical microscopy using trions (charged excitons)10,11,12,13,14,15 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 model16. Our results constitute the first reported real-space observation of quantum liquids in a class of broken symmetry state known as the quantum Hall ferromagnet17,18,19,20,21,22,23,24,25.

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Figure 1: Schematic diagram, transport and optical properties of the sample.
Figure 2: Real-space images of fully spin-polarized (ferromagnetic) FQH liquid around ν = 2/5.
Figure 3: Real-space images of the phase transition between non-magnetic and ferromagnetic FQH liquids around ν = 2/3.
Figure 4: Spatial patterns calculated by the Monte Carlo method using a two-dimensional random field Ising model.


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The authors thank K. Takashina, H. Akiyama, Y. Toda, K. Akiba, N. Kumada, T. D. Rhone, W. J. Munro, T. Fujisawa, N. Shibata, A. Hosoya and H. Sakaki for fruitful discussions and M. Ueki for experimental support. This work was supported by a Grant-in-Aid for Scientific Research (nos. 21241024 and 24241039) from the Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan, and by the Japan Science and Technology Agency, PRESTO. J.H. was supported by a Grant in-Aid from the Tohoku University International Advanced Research and Education Organization.

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J.H. contributed to the experimental set-up and measurements. J.H. and G.Y. contributed to data analysis and numerical calculations. K.M contributed to material growth. G.Y. wrote the manuscript, with feedback from all authors.

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Correspondence to Go Yusa.

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The authors declare no competing financial interests.

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Hayakawa, J., Muraki, K. & Yusa, G. Real-space imaging of fractional quantum Hall liquids. Nature Nanotech 8, 31–35 (2013).

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