Since the realisation that quantum Hall liquids represent topologically ordered phases, the search for new topological states of matter has been a central endeavour in condensed matter physics. Interacting particles in flat band models provide a host of opportunities for creating novel topological phases, baptised fractional Chern insulators (FCI), which are based on realisations of synthetic magnetic fields. First materials realisations have recently been discovered in twisted MoTe₂ bilayers.

The interacting Hofstadter model gives blueprint examples for FCI phases, as predicted by composite fermion theory [1]. We will first review exact diagonalisation results for FCI states stabilised in single Chern bands, and demonstrate that finite-size effects are minimised in the quasi-continuum limit of perfectly flat bands near flux densities n_φ→1/|C| [2,3,4]. We will then discuss a new class of interaction-driven quantum Hall plateau transitions occurring in the Hofstadter model, which arise from the competition of Chern insulator states at weak interaction with FCI states realised at the same particle density for strong interactions. In one such case, our DMRG data at flux density n_φ=3/11 presents a direct transition between a C=4 Chern Insulator and a ν =1/3 Laughlin state, and we examine its exotic critical properties [5]. Even in the non-interacting case, Chern insulators provide new phenomenology in quantum Hall plateau transitions. We will present data on the case of the Haldane model, where the topological C=1 Chern insulator can be destabilised either by increasing disorder or by increasing the effective mass parameter. In our study of the critical behaviour of these two transitions [6], we demonstrate in particular that the mass-driven transition displays critical exponents which vary continuously with the disorder strength.

Finally, going beyond ground state properties of quantum Hall systems, we will expose self-similar features of the dynamical response functions arising for a Laughlin state probed at energies lying above the scale of the single-particle gap [7].

1. Möller, G. & Cooper, N. R. Composite Fermion Theory for Bosonic Quantum Hall States on Lattices. Phys. Rev. Lett. 103, 105303 (2009).
2. Möller, G. & Cooper, N. R. Fractional Chern Insulators in Harper-Hofstadter Bands with Higher Chern Number. Phys. Rev. Lett. 115, 126401 (2015).
3. Andrews, B. & Möller, G. Stability of fractional Chern insulators in the effective continuum limit of Harper-Hofstadter bands with Chern number |C|>1. Phys. Rev. B 97, 035159 (2018).
4. Andrews, B., Neupert, T. & Möller, G. “Stability, phase transitions, and numerical breakdown of fractional Chern insulators in higher Chern bands of the Hofstadter model” Phys. Rev. B 104, 125107 (2021).
5. Schoonderwoerd, L., Pollmann, F. and Möller, G. Interaction-driven plateau transition between integer and fractional Chern Insulators, arxiv:1908.00988 – v2: 2022.
6. 6   J. Mildner, M. D. Caio, G. Möller, N. R. Cooper, and M. J. Bhaseen, Topological Phase Transitions in the Disordered Haldane Model, arxiv:2312.XXXXX (in submission).
7. Andrews, B. & Möller, G. Self-similarity of spectral response functions for fractional quantum Hall states. Proc. R. Soc. A 479, 20230021 (2023).