The Formalism of Conformal Hilbert Spaces and Scale-free Interaction in Fractional Quantum Hall Effect

The fractional quantum Hall (FQH) effect is a family of strongly correlated topological systems in two-dimension, with exotic low lying charge excitations that are anyonic and even non-Abelian. Here we propose a unified framework in understanding the integer and fractional quantum Hall systems via Hilbert space truncation. This framework is closely related to the well-known microscopic pseudopotential and Jack polynomial formalism. The resulting Hilbert spaces with emergent conformal symmetry (i.e. the conformal Hilbert spaces (CHS)) have well-defined topological properties with anyons as "elementary particles". The hierarchical structure of the CHS allows us to reveal internal structures of anyons of the FQH phases (e.g. the Laughlin and Moore-Read phases), and to derive experimentally relevant conditions for such anyons or quasiholes to undergo fractionalisation with the same topological phase. More interestingly, the CHS formalism can also be generalised to the sub-Hilbert spaces of multiple Landau levels (LLs) with scale-free interactions. With such interactions filling factor dependent LL mixing can occur even in the limit of large cyclotron gap. We also propose a novel experimental platform for approximately realising scale-free interactions, that can potentially lead to very robust (non-Abelian) FQH phases from two-body Coulomb-based interaction.

[1] Bo Yang, arXiv: 2307.06361.

[2] Ha Quang Trung and Bo Yang, Phys. Rev. Lett. 127, 046402 (2021).