Classification of ambiguous Chern states

Determine whether the observed ν=1/3 and ν=7/3 states in the θ=1.18° device are integer or fractional states, and resolve whether the ν=1 and ν=3 states in the θ=1.33° device have Chern number magnitude |C|=1 or |C|=2 at zero magnetic field.

Background

The summary table identifies several underdeveloped or ambiguous topological states. For the θ=1.18° device, the measurements cannot distinguish integer from fractional character at ν=1/3 and ν=7/3. For the θ=1.33° device, the zero-field Chern number at ν=1 and ν=3 is difficult to distinguish between magnitude one and two, despite additional finite-field features. These unresolved classifications limit the determination of the complete topology of the observed states.

References

Note that for the ν=\frac{1}{3} and ν=\frac{7}{3} states in the θ=1.18° device, we are not able to distinguish between an integer or fractional state. For the ν=1 and ν=3 states in the θ=1.33° device, it is also difficult to distinguish between |C|=1 or 2 at B=0~T (see Fig.~\ref{fig:S_D2fans} and Fig.~\ref{fig:S_D2VTI}), although we note that at ν=1 there is a clear |C|=2 feature that travels to zero field.

— Incipient superconductivity and tunable Chern insulators in twisted Bernal bilayer-trilayer graphene  (2609.30202 - Waleffe et al., 24 Sep 2026) in Supplementary Materials, Table S1 caption