Layer and spin signatures of spontaneous symmetry breaking in Bernal bilayer graphene

Bernal-stacked bilayer graphene in the presence of an out-of-plane electric displacement field develops a band gap surrounded by flat bands near the band edges with large densities of states. Using capacitance techniques, we study the compressibility of the flat bands at low carrier densities as a function of displacement field. We observe a series of peaks in the compressibility as a function of carrier density that we identify as a sequence of spontaneous isospin symmetry-broken states separated by well-defined phase transitions. Via combined measurements of quantum oscillations, in-plane magnetic fields, and layer-sensitive capacitance techniques, we probe the character of these symmetry-broken states, track the evolution of the phase transitions, and unravel the spin, valley, and layer ordering of each phase. Rhombohedral trilayer graphene also exhibits an analogous sequence of spontaneously symmetry-broken states, though the isospin ordering differs between the two systems.