Visualizing symmetry broken quantum Hall phases and their toplogical boundary modes

The first example of a topological phase of matter is that of integer quantum Hall states, which are characterized by a topological gap in the bulk and the formation of topological boundary modes. Quantum Hall phases with multicomponent electronic degree of freedom, such as valley or layered systems, bring additional richness to the study of topological phases in which the system can also exhibit a broken symmetry. Broadly, these phases of matter are referred to as quantum Hall ferromagnets and have been examined in a number of different contexts. However, much of the work on these system are with macroscopically averaged techniques and the broken symmetry is often inferred from values of Hall conductance. By using the power of STM spectroscopic imaging we have considerably advanced the study of these systems by bringing the ability to visualize the broken symmetry in these systems and to examine their novel boundary modes. 

Our work in this area has focused on a multicomponent quantum Hall phase realized on the surface of bismuth in the presence of a large magnetic field. The valley degeneracy of this system provides a setting in which novel broken symmetry states can emerge in the quantum Hall phase. Our studies in this direction began with our ability to directly image Landau orbits using the STM on the surface of Bi crystals. Using this capability, we were able to directly visualize spontaneous formation of Ising nematic quantum Hall phases in which the electrons occupy one or a combination of valleys (Feldman et al, Science 354, 2016). This valley polarization is driven by an electron-electron interaction and only occurs when Landau orbits are partially filled. Further studies of this system hope to explore the details of valley occupation, discovered that at some occupations the system is in fact quantum Hall ferroelectric (Randeria et al. Nature Physics 14, 2018). Coulomb interactions between electrons have been predicted to drive strongly anisotropic valley systems in this phase — a behavior that our measurements were the first to have confirmed. 

The topological nature of these multicomponent quantum Hall phases require that they host boundary modes. In search of such boundary modes, we uncovered new many-body phenomenon driven by electronic interactions that occur between topological boundary modes at the interface between distinct symmetry-broken states. (Randeria et al. Nature 566, 2019). Using STM spectroscopic imaging, we identified the spontaneous formation of domains of valley-polarized quantum Hall phases and were able to directly examine the properties of their domain wall. At such a domain wall, we directly visualized the closing of the gap and demonstrated the existence of one-dimensional conductance channels, which in this case consisted of counter-propagating modes from different valleys.

In addition to these hallmark signatures of a topological interface, we can further control electrical behavior of the domain wall. By tuning the number and valley “flavor” of the boundary modes, we find the states at the domain wall are gapless (in the case of one pair of counter-propagating modes) or open up a charge gap (for two pairs of counter-propagating modes). In close collaboration with our theory colleagues, we ascribe this striking difference as arising due to an intervalley interaction process only allowed in the latter case, which breaks the valley symmetry protecting the topological boundary modes. Theoretically, these symmetry constrained interaction processes of one-dimensional modes can be mapped to a new class of interacting Luttinger liquids. This work opens exciting avenues to pursue in the future, including the emergence of exotic many-body ground states at different types of topological boundaries and novel electronic applications using the valley degree of freedom.

STM spectroscopic imaging shows domains of different valley polarization on either side of a topological boundary mode (bright line).

STM spectroscopic imaging shows domains of different valley polarization on either side of a topological boundary mode (bright line). The phases are identified by Landau level imaging that reveals the valley polarization by the orientation of the Landau orbits visualized at rare impurities.

Opening and closing a topological gap is detected in STM spectroscopic maps across a domain wall.

Opening and closing a topological gap is detected in STM spectroscopic maps across a domain wall.