Visualizing Landau orbits and anyons in quantum Hall phases

Visualizing Landau Orbits

Our group has directly imaged Landau orbit wavefunctions for the first time in quantum Hall phases using our specially designed ultra-high vacuum, high-field (up to 14T, see facilities section), millikelvin STM system (B. E. Feldman, et al., Science 354, 2016). Our novel experimental approach relies on the fact that the degeneracy of the Landau level (LL) in the quantum Hall states is lifted in the immediate vicinity of a single atomic scale defect. Considering the shape of different degenerate Landau level wavefunctions in the symmetric gauge, for each Nth LL there is an angular momentum state m=N that has a peak at the center. The sharp potential of an atomic scale defect at the center shifted the energy of this state as compared to other angular momentum states. Therefore, mapping the density of states at the center of LL energy in a STM conductance map will result in features that show the absence of an m=N state in the vicinity of atomic scale defects, and allows us to visualize the structure of the Landau orbits spatially. An example of such LL imaging is shown inn Figure 1, where STM conductance maps the N=3 LL wavefunction for the hole band for the two-dimensional electron gas on Bi(111) surface. The number of rings in this image is directly related to the number of nodes of the LL and the spatial extent to the magnetic length, which is tuned with the magnetic field. In these experiments, the shape of the wavefunction is dictated by the anisotropy in the hole pockets band structure as well as the symmetry breaking due to the formation of a valley-polarized quantum Hall system (see section on quantum Hall nematic phases). Theoretical modeling that includes the reduced symmetry of the underlying quantum Hall phase can capture the STM-measured shape of the LL perfectly. Extending the application of LL wavefunction imaging beyond surface states of Bi to other surfaces and 2D systems that are accessible to STM is one of the areas of our current efforts.

Visualizing Anyons

In collaboration with our theoretical colleagues, we have extended the idea of STM imaging of Landau levels in an integer quantum Hall (IQH) state to consider probing fractional quantum Hall (FQH) states (Papić, Z., et al. PRX 2018). These studies reveal that similar measurements near impurities can also reveal the underlying fractional nature of FQH phases and be used to directly probe anyons in this system.

We find that a clear way to distinguish between the local signatures of different FQH states is to consider the spectrum of the localized bound excitation as a function of their angular momentum, m, which would be detected with STM measurements of the local density of states (LDOS) at different radial distances from the center of a single charge impurity. The theoretical energy spectrum of excitations created in the case of electron removal (tunneling electrons out of the sample into the tip) is shown as a function of angular momentum m. The case of the IQH ν=1 shows that tunneling out of each orbital with angular momentum m is shifted down in energy by a different amount because of the gradual shift of the Coulomb potential near the impurity. In this case, there is only one possible state for each m. However, the structure of the LDOS near the impurity is different in FQH states. For ν=1/3, as shown in Figure 2, rather than one state per m, most of the m’s have multiple energies appearing in the LDOS. Although some of these states belong to a high-energy continuum, there is also a band of discrete low-energy levels whose multiplicity grows with angular momentum. This is a signature of fractionalization since, when an electron is removed from the ν=1/3 state, the hole splits into three charged e/3 quasiholes, and there are several three-quasihole states for a given total angular momentum ℏm. The fractional statistics changes the angular momentum, and hence the counting of these states. Thus, in a FQH phase, where the hole splits into q charge e/q anyons, the orbitally resolved LDOS reveals all the distinct eigenstates in which the q anyons have total angular momentum ℏm. For many FQH states the observed multiplicities can be predicted using “exclusion statistics.” Hence, the impurity LDOS provides a “fingerprint” for identifying the topological order. Even the presence of a trapped anyon can be visualized using this method.

The orbital resolved LDOS

Figure 2. The orbital-resolved LDOS. (a) The IQH state, one level for each m. (b) The v=1/3 FQH states, multiple energy levels for each m. (c) The v=5/2 Moore-Read state. (d) same as (b) but with a charge e/3 anyon bound to the impurity.