Probing symmetry breaking of the fractionally filled Landau level with a scanning tunneling microscope*

In monolayer graphene, the spin and valley degeneracy lead to a fourfold isospin degeneracy of the macroscopically degenerate Landau levels. The partially filled landau level in particular hosts a myriad of exotic phases such as the fractional quantum hall states[1], Wigner crystals and skyrmion solids[2]. Understanding the spin and valley symmetries of the electronic wavefunctions is thus crucial in characterizing the ground states and excitations. Exploiting the valley-sublattice equivalence in the zeroth Landau level (zLL), we previously employed gate-tunable scanning tunneling microscopy and spectroscopy (STM-STS) to quantify the valley symmetry of the ground state at integer fillings of the zLL [3], whereas the spectral features at fractional fillings were more susceptible to the tip-sample work function mismatch. In this work, we show that through careful characterization of the spectral features we can reproducibly make STM tips that have minimal effect on the electronic states at partial fillings, enabling us to underpin the valley order of the fractionally filled zLL in a pristine monolayer graphene device. We resolve the spectral features attributed to Haldane pseudopotentials[4], showing remarkable agreement with the exact diagonalization calculations. Moreover, we detect a magnetic field dependent isospin transition near nu = -2/3 marked by a sharp decrease in sublattice polarization of the ground state. We then probe the energy dependence of the valley polarization in fractional excitations beyond the theoretically accessible regime which exhibits excellent particle-hole symmetry. Our findings establish STS as a robust technique to probe the intricate charge ordering of highly interacting electron systems. 

[1] C. R . Dean et al., Nat. Phys. 7, 693–696 (2011)

[2] H. Zhou et al., Nat. Phys. 16, 154–158 (2020)

[3] X. Liu et al., Science 375, 321-326 (2021)

[4] A. H. MacDonald, Phys. Rev. Lett. 105, 206801 (2010)