  Nontrivial interacting phases can emerge in elementary materials. As a prime
example, continuing advances in device quality have facilitated the observation
of a variety of spontaneous quantum Hall-like states, a cascade of Stoner-like
magnets, and an unconventional superconductor in bilayer graphene. Its natural
extension, rhombohedral trilayer graphene is predicted to be even more
susceptible to interactions given its even flatter low-energy bands and larger
winding number. Theoretically, five spontaneous quantum Hall phases have been
proposed to be candidate ground states. Here, we provide transport evidence for
observing four of the five competing ordered states in interaction-maximized,
dually-gated, rhombohedral trilayer graphene. In particular, at vanishing but
finite magnetic fields, two states with Chern numbers 3 and 6 can be stabilized
at elevated and low electric fields, respectively, and both exhibit clear
magnetic hysteresis. We also reveal that the quantum Hall ferromagnets of the
zeroth Landau level are ferroelectrics with spontaneous layer polarizations
even at zero electric field, as evidenced by electric hysteresis. Our findings
exemplify the possible birth of rich interacting electron physics in a simple
elementary material.
