  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.

Mesoscale and Nanoscale Physics