Butterflies from Information Metric

We study time evolution of distance between thermal states excited by local operators, with different external couplings. We find that growth of the distance implies growth of commutators of operators, signifying the local excitations are scrambled. We confirm this growth of distance by holographic computation, by evaluating volume of codimension 1 extremal volume surface. We find that the distance increases exponentially as $e^{\frac{2\pi t}{\beta}}$. Our result implies that, in chaotic system, trajectories of excited thermal states exhibit high sensitivity to perturbation to the Hamiltonian, and the distance between them will be significant at the scrambling time. We also confirm the decay of two point function of holographic Wilson loops on thermofield double state.