The study of interacting quantum Hall states and their exotic anyonic excitations poses a major challenge in current experimental and theoretical research. Quantum simulators, in particular ultracold atoms, provide a promising platform to realize, manipulate, and understand such systems with a high controllability. In many cases, the experimental platforms use lattices, so that a theoretical understanding of the connection between continuum quantum Hall states and their lattice analogs - called Chern insulators - is desirable. A near-term goal for theory is to find suitable observables to directly probe the topological properties of exotic states of matter. To this end, I will propose new directions available in state-of-the-art experiments, focusing specifically on two examples. First, I will present results for the Laughlin state showing that at magnetic filling factor $\nu=1/2$ three-legged systems are sufficient to realize and probe properties of central interest in interacting topological systems, like the central charge [1]. I will also address the hidden order in FQH states and how it can be detected with snapshots obtained via perfect sampling from an MPS [2]. Afterwards, I will discuss a bosonic analog of the Pfaffian state at filling factor $ \nu=1$, which was long studied as a candidate state for the fermionic $\nu=5/2$ plateau and may provide a route towards non-Abelian anyon braiding [3].
[1] Palm et al., Snapshot-based detection of 𝜈=1/2 Laughlin states: Coupled chains and central charge; PRB 106 (2022) [2] Pauw et al., Detecting Hidden Order in Fractional Chern Insulators; arXiv:2309.03666. [3] Palm et al., Bosonic Pfaffian state in the Hofstadter-Bose-Hubbard model; PRB 103 (2021)