Direct measurement of discrete valley and orbital quantum numbers in a multicomponent quantum Hall system

Strongly interacting two dimensional electron systems (2DESs) host a complex landscape of broken symmetry states. The possible ground states are further expanded by internal degrees of freedom such as spin or valley-isospin. While direct probes of spin in 2DESs were demonstrated two decades ago, the valley quantum number has only been probed indirectly in semiconductor quantum wells, graphene mono- and bilayers, and transition-metal dichalcogenides. Here, we present the first direct experimental measurement of valley polarization in a two dimensional electron system, effected via the direct mapping of the valley quantum number onto the layer polarization in bilayer graphene at high magnetic fields. We find that the layer polarization evolves in discrete steps across 32 electric field-tuned phase transitions between states of different valley, spin, and orbital polarization. Our data can be fit by a model that captures both single particle and interaction induced orbital, valley, and spin anisotropies, providing the most complete model of this complex system to date. Among the newly discovered phases are theoretically unanticipated orbitally polarized states stabilized by skew interlayer hopping. The resulting roadmap to symmetry breaking in bilayer graphene paves the way for deterministic engineering of fractional quantum Hall states, while our layer-resolved technique is readily extendable to other two dimensional materials where layer polarization maps to the valley or spin quantum numbers, providing an essential direct probe that is a prerequisite for manipulating these new quantum degrees of freedom.
Comment: This version (v3) updates a large portion of the theoretical treatment in previous versions and consequently has new Figs. 3 and 4 of the main text and a significant overhaul of the Supplementary Material