Three-layer coherent quantum Hall effect

The \nu = 1 quantum Hall state in closely spaced double layer system is demonstrated to be layer-coherent, i.e. electrons in two layers behave as if they are in one, which leads to the superfluid of indirect excitons. In this work, we report for the first time a triple-layer system with two hBN tunnel barriers between three graphene sheets. The combination of voltage bias on the top, bottom gate electrodes and on the bottom graphene renders the independent control the carrier density in each layer. By mapping out the total compressibility of all three graphene layers in the three-dimensional density space, we observe incompressible quantum Hall states on the two-dimensional plane of total filling fraction \nu = 1 and 2. The robustness of this quantum Hall state against charge distribution among all three layers suggests phase coherence across all three layers. The energy gap as a function of filling fraction in each layer matches well with the expectation from theory. Incompressible states at total filling fraction \nu = 4/3 and 5/3 are also observed in part of the density space, hinting at a fractional three-layer coherent state.