L\'evy Light Cones and Critical Causality in Fractional Multiscale Quantum Ising Models

We study causality and criticality in a one-dimensional fractional multiscale transverse-field Ising model, where fractional derivatives generate long range interactions beyond the scope of standard power laws. Such fractional responses are common in classical systems including the anomalous stress-strain behaviour of viscoelastic polymers, L\'evy-like contaminant transport in heterogeneous porous media, and the non-Debye dielectric relaxation of glassy dielectrics. Furthermore, these unique interactions can be implemented in current quantum information architectures, with intriguing consequences for the many-body dynamics. Using a truncated Jordan-Wigner approach, we show that in the long wavelength limit of the mean field, the dynamical critical exponent is set by the fractional order q as $z=q/2$. To probe genuine many-body dynamics, we apply matrix-product-state simulations with the time-dependent variational principle adapted to nonlocal couplings. Tracking the entanglement-entropy light cone and performing finite-size scaling of the many-body gap for $0<q<2.5$, we confirm a continuously tunable exponent $z(q)$: for $q<2$ the entanglement front broadens with a sublinear light cone; for $2<q<2.5$ we observe a faint superlinear cone indicative of $z<1$; and for $q \gtrsim 2.5$ the system reverts to the ballistic nearest-neighbour regime with $z=1$. The correspondence between quantum entanglement fronts that spread as $t^{1/z}$ and classical L\'evy flights whose mean-square displacement grows as $t^{2/q}$ provides a direct physical link between fractional interactions and L\'evy statistics. Fractional derivatives therefore offer a unified framework in which short-range, power-law, and frustrated long-range interactions emerge as limiting cases, enabling controlled exploration of nonlocal causality bounds and exotic entanglement dynamics within current quantum information platforms.