Three-Dimensional Non-Foliated Fractional Quantum Hall Phases with Irrational Anyons in Twisted van der Waals Multilayers

Abstract

Three-dimensional fractional quantum Hall phases offer a route to intrinsically higher-dimensional topological order beyond simple stacks of two-dimensional quantum Hall liquids. Such phases can exhibit non-foliated, intrinsically three-dimensional entanglement structures, exponentially large topological degeneracies and quasiparticles with irrational braiding statistics. Their microscopic realization has remained elusive because Landau quantization in three dimensions generally leaves dispersive one-dimensional bands, favoring metallic and density-wave states over incompressible fractional liquids. Here we show that large-angle twisted van der Waals multilayers provide a practical route around this obstruction. Large twist angles suppress coherent interlayer tunneling through momentum mismatch, while the atomic-scale layer separation preserves strong interlayer Coulomb interactions. Using Monte Carlo calculations to compare the energies of an extensive set of 862 competing trial wavefunctions, we find that generalized Halperin states with quantum coherence extending across multiple consecutive layers are stabilized. In experimentally accessible magnetic-field regimes, these states replace the metallic spontaneous-interlayer-coherent phases that dominate conventional untwisted graphite-like multilayers. The resulting liquids realize non-foliated fractional quantum Hall order closely related to fractonic topological order, hosting quasiparticles with rational electric charges but irrational braiding statistics. Their large topological degeneracy and non-rational statistical phases may offer unconventional resources for quantum information storage and processing. Our results establish twisted van der Waals multilayers as a realistic materials platform for three-dimensional fractional Hall matter beyond conventional two-dimensional quantum Hall systems.