  In view of the evolution from the integer to fractional quantum Hall effect,
the next frontier in the research of topological insulators is to investigate
what happens in fractionally filled topological flat bands. A particularly
pressing question is if there exists the lattice analogue of the Laughlin state
in the 1/3-filled Chern flat band, dubbed as the Chern-Laughlin state. The
answer depends crucially on the form of the electron-electron interaction,
which can generate various competing ground states such as the Laughlin,
stripe/nematic, parafermion, and parton states. Unfortunately, it is difficult
to precisely characterize the exact ground state as any of these candidate
ground states due to the lack of appropriate order parameters. Here, we propose
that the existence of an adiabatic path from fractional Chern insulators to the
Tao-Thouless state, i.e., the root partition state of the Laughlin state in the
thin torus limit, can serve as an effective order parameter for the
Chern-Laughlin state. Specifically, by devising the piecewise hybrid adiabatic
path of first transforming the electron-electron interaction and then taking
the thin torus limit, it is shown that Chern flat bands with the
nearest-neighbor interaction can indeed host the Chern-Laughlin state at 1/3
filling. This method can be extended to possible FCIs at other general fillings
of the Jain sequence.
