Abstract
We explore a model of free fermions in one dimension, subject to frustrated (non-commuting) local measurements across adjacent sites, which resolves the fermions into non-orthogonal orbitals, misaligned from the underlying lattice. For maximal misalignment, superdiffusive behavior emerges from the vanishing of the measurement-induced quasiparticle decay rate at one point in the Brillouin zone, which generates fractal-scaling entanglement entropy S ∝ ℓ^1/3 for a subsystem of length ℓ. We derive an effective non-linear sigma model with long-range couplings responsible for L\'evy flights in entanglement propagation, which we confirm with large-scale numerical simulations. When the misalignment is reduced, the entanglement exhibits, with increasing ℓ, consecutive regimes of superdiffusive, S∝ ℓ^1/3, diffusive, S∝ ln ℓ, and localized, S = const, behavior. Our findings show how intricate fractal-scaling entanglement can be produced for local Hamiltonians and measurements.