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Novel broken symmetry states can spontaneously form due to Coulombinteractions in electronic systems with multiple internal degrees of freedom.Multi-valley materials offer an especially rich setting for the emergence ofsuch states, which have potential electronic and optical applications. To date,identification of these broken symmetry phases has mostly relied on theexamination of macroscopic transport or optical properties. Here we demonstratea powerful direct approach by visualizing the wave functions of bismuth surfacestates with a scanning tunneling microscope. Strong spin-orbit coupling on thesurface of bismuth leads to six degenerate teardrop-shaped hole pockets. Ourspectroscopic measurements reveal that this degeneracy is fully lifted at highmagnetic field as a result of exchange interactions, and we are able todetermine the nature of valley ordering by imaging the broken symmetry Landaulevel (LL) wave functions. The remarkable spatial features of singly degenerateLL wave functions near isolated defects contain unique signatures ofinterference between spin-textured valleys, which identify the electronicground state as a quantum Hall ferroelectric. Our observations confirm therecent prediction that interactions in strongly anisotropic valley systemsfavor the occupation of a single valley, giving rise to emergentferroelectricity in the surface state of bismuth.

Ferroelectric quantum Hall phase revealed by visualizing Landau level wavefunction interference