Quantum parity Hall effect in Bernal-stacked trilayer graphene

The quantum Hall effect has recently been generalized from transport of conserved charges to include transport of other approximately conserved-state variables, including spin and valley, via spin- or valley-polarized boundary states with different chiralities. Here, we report a class of quantum Hall effect in Bernal- or ABA-stacked trilayer graphene (TLG), the quantum parity Hall (QPH) effect, in which boundary channels are distinguished by even or odd parity under the system's mirror reflection symmetry. At the charge neutrality point, the longitudinal conductance [Formula: see text] is first quantized to [Formula: see text] at a small perpendicular magnetic field [Formula: see text], establishing the presence of four edge channels. As [Formula: see text] increases, [Formula: see text] first decreases to [Formula: see text], indicating spin-polarized counterpropagating edge states, and then, to approximately zero. These behaviors arise from level crossings between even- and odd-parity bulk Landau levels driven by exchange interactions with the underlying Fermi sea, which favor an ordinary insulator ground state in the strong [Formula: see text] limit and a spin-polarized state at intermediate fields. The transitions between spin-polarized and -unpolarized states can be tuned by varying Zeeman energy. Our findings demonstrate a topological phase that is protected by a gate-controllable symmetry and sensitive to Coulomb interactions.