Universal Growth of Krylov Complexity Across a Quantum Phase Transition

Abstract

We study the statistical properties of the spread complexity in the Krylov space of quantum systems driven across a quantum phase transition. Using the diabatic Magnus expansion, we map the evolution to an effective one-dimensional hopping model. For the transverse field Ising model, we establish an exact link between the growth of complexity and the Kibble-Zurek defect scaling: all cumulants of complexity exhibit the same power-law scaling as the defect density, with coefficients identical to the mean, and the full distribution asymptotically becomes Gaussian. We also provide a general scaling argument for the complexity growth across arbitrary second-order quantum phase transitions, which is further demonstrated numerically in the long-range Kitaev models, both for short and long-range couplings.