Measurement-Induced Lévy Flights of Quantum Information

We explore a model of free fermions in one dimension, subject to (noncommuting) local measurements across adjacent sites, which resolves the fermions into nonorthogonal 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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mrow><a:mi>S</a:mi><a:mo>∝</a:mo><a:msup><a:mrow><a:mo>ℓ</a:mo></a:mrow><a:mrow><a:mn>1</a:mn><a:mo>/</a:mo><a:mn>3</a:mn></a:mrow></a:msup></a:mrow></a:math> for a subsystem of length <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mrow><c:mo>ℓ</c:mo></c:mrow></c:math>. We derive an effective nonlinear sigma model with long-range couplings responsible for Lévy flights in entanglement propagation, which we confirm with large-scale numerical simulations. When the misalignment is reduced, the entanglement exhibits, with increasing <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mo>ℓ</e:mo></e:math>, consecutive regimes of superdiffusive, <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mrow><g:mi>S</g:mi><g:mo>∝</g:mo><g:msup><g:mrow><g:mo>ℓ</g:mo></g:mrow><g:mrow><g:mn>1</g:mn><g:mo>/</g:mo><g:mn>3</g:mn></g:mrow></g:msup></g:mrow></g:math>, diffusive, <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mrow><i:mi>S</i:mi><i:mo>∝</i:mo><i:mi>ln</i:mi><i:mo>ℓ</i:mo></i:mrow></i:math>, and localized, <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mi>S</k:mi><k:mo>=</k:mo><k:mtext>const</k:mtext></k:math>, behavior. Our findings show how intricate fractal-scaling entanglement can be produced for local Hamiltonians and measurements.