Does scrambling equal chaos?

TL;DR

It is shown that the parametrically long exponential growth of out-of-time order correlators (OTOCs), also known as scrambling, does not necessitate chaos, and a lower bound on the OTOC Lyapunov exponent is derived, which depends only on local properties of such fixed points.

Abstract

Focusing on semiclassical systems, we show that the parametrically long exponential growth of out-of-time order correlators (OTOCs), also known as scrambling, does not necessitate chaos. Indeed, scrambling can simply result from the presence of unstable fixed points in phase space, even in a classically integrable model. We derive a lower bound on the OTOC Lyapunov exponent which depends only on local properties of such fixed points. We present several models for which this bound is tight, i.e. for which scrambling is dominated by the local dynamics around the fixed points. We propose that the notion of scrambling be distinguished from that of chaos.

Figures (5)

• Figure 1: (a) Extended exponential growth of the infinite-temperature OTOC \ref{['eq:OTOC']} of the integrable LMG model \ref{['eq:H_classical']} in the semi-classical limit. The growth saturates at the Ehrenfest time $\sim \ln (S)$. The exponent $\lambda_{\text{OTOC}} = \sqrt{3}$ is the unstable exponent of the saddle point in the classical phase space. (b) Microcanonical-ensemble OTOCs $C_E(t)=-\frac{1}{125}\sum_{\epsilon \in b_E}\braket{\epsilon|[\hat{S}_z, \hat{S}_z(t)]^2|\epsilon}$ ($S = 2500$), where $b_E$ is an energy window of $125$ Hamiltonian eigenstates $\{\vert \epsilon \rangle \}$ with average energy $E$. A few representative ensembles across the entire energy spectrum are shown. The one with $E \approx 1$, corresponding to the classical saddle, dominates the exponential growth observed in (a). Inset: Energy landscape of the classical limit, with the same color code as (b), and the saddle in the center.
• Figure 2: The markers show the instantaneous exponential growth rate $\ln [C(t+1)/C(t)]$ as a function of the number of kicks $t$ in the kicked rotor model, quantized with Planck constant $\hbar_{\text{eff}} = 2^{-14}$ (see Ref. galitski for definition and methods). Good agreement can be seen with the saddle point exponents $\omega(K)$ from Eq. \ref{['eq:omega_rotor']}, plotted as horizontal lines.
• Figure 3: (a-c) Growth of OTOC \ref{['eq:OTOC']} where $\hat{O} = \hat{x}_1 + \hat{x}_2$, in the Feingold-Peres model quantized to $S = 75$, and $c=-0.6, 0,$ and $0.9$. The dashed lines are straight lines with slope given by $\omega(c)$ from Eq. \ref{['eq:lowerbound']}; for $c = 0.9$, $\omega(c) = 0$, and the OTOC is oscillatory. (d) The data points represent the exponent $\lambda_{\text{OTOC}}$ extracted from the growth of $C(t)$. The continuous curve is $\omega(c)$ from Eq. \ref{['eq:lowerbound']}.
• Figure 4: Exponents of the saddle point ($\lambda_{\text{saddle}}$) vs chaos ($2 \lambda_{\text{chaos}}$) contributions to the OTOC in the mean-field depinning model \ref{['eq:Fisher']} ($\sigma = 2$ and $N = 128$). In each case, we cool down the system from $T = 2$ gradually to $T= 0.05$, generating along the way $10000$ configurations at different energy densities. To estimate the chaos contribution, we compute the sensitivity up to $t = 50$ starting from each configuration and extract $\lambda_{\text{chaos}}$. We plot the resulting $(E/N, 2\lambda_{\text{chaos}})$ as red crosses. To estimate the saddle contributions, we perform gradient descent from each configuration to reach an equilibrium, and compute its $\lambda_{\text{saddle}}$. The positive values are plotted as dark dots. At low energies, $\lambda_{\text{chaos}}$ is severely suppressed, while saddles with large contribution to OTOC are still abundant.
• Figure 5: (a) The OTOC in the quantum Dicke model ($N = 40$) with $\hat{O} = \hat{p}$, for a microcanonical ensemble of 40 eigenstates around $H = -1$, and for two representative values of $\gamma$. The dashed slopes show $\lambda_\text{sad}=\omega_1 = \sqrt{\gamma - 1}$. (b) Classical Lyapunov exponent in the Dicke model with $\gamma=2$ (computed as $(\partial q(t) / \partial q(0))^2 \sim e^{2 \lambda_{\text{chaos}} t}$, with $t = 2000$) for $600$ randomly sampled trajectories in the energy shell $[-1.3, 0.7]$. For all of them, $\lambda_{\text{chaos}}$ is smaller than $\lambda_{\text{saddle}}=\omega_1 = 1$, marked by a star.