  Novel broken symmetry states can spontaneously form due to Coulomb
interactions in electronic systems with multiple internal degrees of freedom.
Multi-valley materials offer an especially rich setting for the emergence of
such states, which have potential electronic and optical applications. To date,
identification of these broken symmetry phases has mostly relied on the
examination of macroscopic transport or optical properties. Here we demonstrate
a powerful direct approach by visualizing the wave functions of bismuth surface
states with a scanning tunneling microscope. Strong spin-orbit coupling on the
surface of bismuth leads to six degenerate teardrop-shaped hole pockets. Our
spectroscopic measurements reveal that this degeneracy is fully lifted at high
magnetic field as a result of exchange interactions, and we are able to
determine the nature of valley ordering by imaging the broken symmetry Landau
level (LL) wave functions. The remarkable spatial features of singly degenerate
LL wave functions 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 prediction that interactions in strongly anisotropic valley systems
favor the occupation of a single valley, giving rise to emergent
ferroelectricity in the surface state of bismuth.
