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Ferroelectric quantum Hall phase revealed by visualizing Landau level wavefunction interference


States with spontaneously broken symmetry can form due to Coulomb interactions in electronic systems with multiple internal degrees of freedom. Materials with several degenerate regions in the Brillouin zone—called valleys—offer a rich setting for the emergence of such states, which have potential electronic and optical applications1,2,3,4. To date, identification of these broken-symmetry phases has mostly relied on macroscopic transport or optical properties. Here we demonstrate a direct approach by visualizing the wavefunctions of bismuth surface states with a scanning tunnelling microscope. Strong spin–orbit coupling on the surface of bismuth leads to six degenerate, teardrop-shaped, hole valleys5. Our spectroscopic measurements reveal that exchange interactions fully lift this degeneracy at high magnetic field, and we are able to determine the nature of the valley ordering by imaging the broken-symmetry Landau level wavefunctions. The spatial features of singly degenerate Landau level wavefunctions near isolated defects contain unique signatures of interference between spin-textured valleys, which identify the electronic ground state as a quantum Hall ferroelectric. Our observations confirm the recent prediction6 that interactions in strongly anisotropic valley systems favour the occupation of a single valley, giving rise to emergent ferroelectricity in the surface state of bismuth.

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Fig. 1: Exchange splitting and broken-symmetry states at odd-integer LL filling.
Fig. 2: Imaging LL wavefunctions at odd-integer filling factors.
Fig. 3: Interference fringes around a surface defect due to mixing of valley-polarized states.
Fig. 4: Diagonal interference patterns reflecting the nodal wavefunction character and spin texture of the pockets.


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We would like to thank I. Sodemann and L. Fu for helpful discussions. Work at Princeton has been supported by the Gordon and Betty Moore Foundation as part of the EPiQS initiative (GBMF4530), DOE-BES grant DE-FG02-07ER46419, the ARO-MURI program W911NF-12-1-046, NSF-MRSEC programs through the Princeton Center for Complex Materials DMR-142054 and NSF-DMR-1608848, the Eric and Wendy Schmidt Transformative Technology Fund at Princeton, an NSF Graduate Research Fellowship (M.T.R.) and a Dicke fellowship (B.E.F.). Work at Austin was supported by DOE grant DE-FG03-02ER45958 and by the Welch Foundation grant TBF1473. The work of F.W. at Argonne is supported by the Department of Energy, Office of Basic Energy Science, Materials Science and Engineering Division.

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M.T.R., B.E.F., H.D., A.G. and A.Y. designed and conducted the STM measurements and their analysis. F.W. and A.H.M. performed the theoretical modelling and simulations. H.J. and R.J.C. synthesized the samples. All authors contributed to the writing of the manuscript.

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Correspondence to Ali Yazdani.

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Supplementary Information

Supplementary figures 1–10, theoretical details and numerical simulations

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Randeria, M.T., Feldman, B.E., Wu, F. et al. Ferroelectric quantum Hall phase revealed by visualizing Landau level wavefunction interference. Nature Phys 14, 796–800 (2018).

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