Butterflies from Information Metric

TL;DR

This work develops a quantitative, information-theoretic probe of scrambling in chaotic quantum systems by examining the Fisher information metric $G_{\lambda\lambda}$ on thermofield double states perturbed by local operators. The authors show that $G_{\lambda\lambda}$ naturally decomposes into a constant, perturbation-independent part and a scrambling part driven by growing operator commutators, with the chaotic large-$N$ dynamics causing the scrambling term to scale as $\sim e^{\frac{2\pi}{\beta}t}$. Using AdS/CFT, they holographically compute the growth via the extremal-volume surface $\mathrm{Vol}(\Sigma)$ in shock-wave geometries, obtaining $|\langle\Psi_{TFD}(\lambda+\delta\lambda)|\Psi_{TFD}(\lambda)\rangle|=e^{-n_d(\delta\lambda)^2\mathrm{Vol}(\Sigma)}$ and showing linear small-$\alpha$ growth $\Delta\mathrm{Vol}=f_1\alpha$ with $\alpha\propto e^{\frac{2\pi}{\beta}t_w}$, hence $G_{\lambda\lambda}^{W:c}=f_1\alpha$. They also confirm consistent decay of Wilson loop correlators $\langle W_LW_R\rangle\sim e^{-(S_{NG}-\text{reg})}$, reinforcing the connection between information-theoretic distance and scrambling. Overall, the paper connects the butterfly effect, chaos bounds, and holographic complexity-like bulk volumes to provide a concrete, calculable measure of scrambling in thermal states.

Abstract

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 $e^{\frac{2πt}β}$. 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.

Figures (3)

• Figure 1: Orbits of states with slightly different Hamiltonians. For chaotic theory or chaotic perturbation to the original Hamiltonian, the distance between states grows exponentially even if original states are the same or close to each other.
• Figure 2: A diagram of the shock wave geometry. Black line are boundaries, dashed lines are singularities, and a blue line is the shock wave. Red surface connecting two boundaries is the extremal volume surface $\Sigma$. For 2d CFT, this surface can be understood as configuration of world sheet of a string ending on both boundaries.
• Figure 3: Numerical plot for growth of information metric versus $\alpha={E\over 4M}e^{{2\pi\over \beta}t_w}$ in $AdS_3/CFT_2$. The numbers in the right are radiuses of BH horizon $R={2\pi\over \beta}$.The vertical ax is for ${2\Delta{\rm Vol}\over R^2}$, and horizontal ax is for $\alpha$. We can confirm linear growth of information $\Delta{\rm Vol}=f_1\times \alpha+\mathcal{O}(\alpha^2)$ for small $\alpha$.