Measuring the distance between quantum many-body wave functions

We study the distance of two wave functions under chaotic time evolution. The two initial states differ only by a local perturbation. To be entitled 'chaos' the distance should have a rapid growth afterwards. Instead of focusing on the entire wave function, we measure the distance d(2)(t) by investigating the difference of two reduced density matrices of the subsystem A that is spatially separated from the local perturbation. This distance d(2)(t) grows with time and eventually saturates to a small constant. We interpret the distance growth in terms of operator scrambling picture, which relates d(2)(t) to the square of commutator C(t) (out-of-time-order correlator) and shows that both these quantities measure the area of the operator wave front in subsystem A. Among various one-dimensional spin-1/2 models, we numerically show that the models with non-local power-law interaction can have an exponentially growing regime in d(2)(t) when the local perturbation and subsystem A are well separated. This so-called Lyapunov regime is absent in the spin-1/2 chain with local interaction only. After sufficiently long time evolution, d(2)(t) relaxes to a small constant, which decays exponentially as we increase the system size and is consistent with eigenstate thermalization hypothesis. Based on these results, we demonstrate that d(2)(t) is a useful quantity to characterize both quantum chaos and quantum thermalization in many-body wave functions.