Visualizing the fractional topological order: From fractional Chern insulators to the Tao-Thouless state

Strong correlation and topology are two main pillars of modern physics, which can be bridged by fractional quantum Hall states and their lattice analogs known as fractional Chern insulators. A particularly pressing question for fractional Chern insulators is if there exists a lattice analog of the Laughlin state in the 1/3-filled Chern flat band dubbed the Chern-Laughlin state, and if so, how to detect it experimentally. The exact ground state in the 1/3-filled Chern flat band depends on the form of the electron-electron interaction, which can generate various competing ground states derived from, for example, the Laughlin, parafermion, parton, and stripe/nematic states in the 1/3-filled Landau level. In this context, it is of critical importance to precisely identify the exact ground state for a given electron-electron interaction. Here, we propose that the existence of an adiabatic path from the 1/3-filled fractional Chern insulator 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 precise identification of the Chern-Laughlin state. Specifically, by devising a hybrid adiabatic path of first deforming the electron-electron interaction and then taking the thin torus limit, we show that two-dimensional bulk Chern flat bands can host the Chern-Laughlin state at 1/3 filling. More importantly, our results suggest that, unless the electron-electron interaction is strictly of the nearest-neighbor form, the Tao-Thouless state can actually be detected in experiments, for example by using an incommensurate double-walled carbon nanotube made out of the magic-angle twisted bilayer graphene, which allows charge-sensitive surface measurements such as scanning tunneling microscopy. This method can be extended to various other fractional Chern insulators corresponding to both Abelian and non-Abelian fractional quantum Hall states.