Fractonic Fractional Quantum Hall Effect

In non-interacting systems, disorder can drive a trivial phase into a topological one. However little is known how to construct a fractional quantum Hall ground-state, a paradigmatic topologically ordered state, that exists both in crystalline and disordered lattices and is qualitatively different to known topological phases. Here, we propose a general method for building such a phase. This is done by coupling quantum wires placed aperiodically in real-space, where the spatial positioning allows us to tune the inter-wire couplings. We call the emergent phase the Fractonic Fractional Quantum Hall Effect as it displays a rich interplay of fractional quantum Hall physics with fractonic constraints, formed by coupling differently-fractionalised wires into a globally gapped phase. The ground state has an exponential degeneracy in system size, a signature of the emergence of fractons. It displays a rich phenomenology of excitations, which can either behave like anyons confined to move in one dimension (lineons), multiples of which can then hop between two wires (s-lineons) or be free to travel across the system (C-anyons), depending on the multiplicity. Both the ground state degeneracy and mutual statistics are directly determined by the real-space positions of the wires, which can be disordered. Our method provides an analytically solvable pathway to non-crystalline fractional quantum Hall effects and fractonic theories in two-dimensions, examples of which were lacking.