  A two-dimensional electron system placed in a magnetic field develops Landau
levels, where strong Coulomb interactions lead to the appearance of many-body
correlated ground states. Quantum numbers similar to the electron spin enable
the understanding and control of complex ground state order and collective
excitations. Owing to its spin, valley and orbital degrees of freedom,
Bernal-stacked bilayer graphene offers a rich platform to pursue correlated
phenomena in two dimensions. In this work, we fabricate dual-gated
Bernal-stacked bilayer graphene devices and demonstrate unprecedented fine
control over its valley isospin degrees of freedom using a perpendicular
electric field. Higher sample quality enables us to probe regimes obscured by
disorder in previous studies. We present evidence for a new even-denominator
fractional quantum Hall state at filling factor {\nu} = 5/2. The 5/2 state is
found to be spontaneously valley polarized in the limit of vanishing valley
Zeeman splitting, consistent with a theoretical prediction made regarding the
spin polarization of the Moore-Read state. In the vicinity of the
even-denominator fractional quantum Hall states, we observe the appearance of
the predicted Levin-Halperin daughter states of the Moore-Read Pfaffian wave
function at {\nu}= 3/2, 7/2, and -1/2 and of the anti-Pfaffian at {\nu}= 5/2.
These observations suggest the breaking of particle-hole symmetry in bilayer
graphene. We construct a comprehensive valley polarization phase diagram for
the Jain sequence fractional states surrounding filling factor 3/2. These
results are well explained by a two-component composite fermion model, further
demonstrating the SU(2) nature of the valley isospin in bilayer graphene. Our
experiment paves the path for future efforts of manipulating the valley isospin
in bilayer graphene to engineer exotic topological orders and quantum
information processes.
