  Graphene and its multilayers have attracted considerable interest owing to
the fourfold spin and valley degeneracy of their charge carriers, which enables
the formation of a rich variety of broken-symmetry states and raises the
prospect of controlled phase transitions among them. In especially clean
samples, electron-electron interactions were recently shown to produce
surprising patterns of symmetry breaking and phase transitions in the integer
quantum Hall regime. Although a series of robust fractional quantum Hall states
was also recently observed in graphene, their rich phase diagram and tunability
have yet to be fully explored. Here we report local electronic compressibility
measurements of ultraclean suspended graphene that reveal a multitude of
fractional quantum Hall states surrounding filling factors v = -1/2 and -1/4.
In several of these states, we observe phase transitions that indicate abrupt
changes in the underlying order and are marked by a narrow region of negative
compressibility that cuts across the incompressible peak. Remarkably, as
filling factor approaches v = -1/2, we observe additional oscillations in
compressibility that appear to be related to the phase transitions and persist
to within 2.5% of v = -1/2. We use a simple model based on crossing Landau
levels of composite particles with different internal degrees of freedom to
explain many qualitative features of the experimental data. Our results add to
the diverse array of correlated states observed in graphene and demonstrate
substantial control over their order parameters, showing that graphene serves
as an excellent platform to study correlated electron phases of matter.
