Exponential growth of out-of-time-order correlator without chaos: inverted harmonic oscillator

TL;DR

This work shows that thermal out-of-time-order correlators can exhibit exponential growth in time even in classically non-chaotic, one-dimensional quantum systems with an inverted-harmonic-oscillator saddle. By solving the 1D Schrödinger equation for both soft-wall and hard-wall potentials and evaluating microcanonical and thermal OTOCs, the authors demonstrate that the growth is governed by the saddle's classical Lyapunov exponent and persists at high temperatures, yielding a finite $\lambda_{OTOC}$ that scales as $\mathcal{O}(\lambda_{saddle})$. They establish universality across potential shapes and derive a one-dimensional bound, $\lambda_{OTOC}(T) \lesssim c\,T$ with $c=\mathcal{O}(1)$, by invoking energy and resolution constraints for probing the hilltop. The findings challenge the notion that exponential OTOC growth is exclusively tied to chaos, and they connect to broader themes in chaos bounds and holography, suggesting scrambling can occur without chaos in simple quantum systems.

Abstract

We provide a detailed examination of a thermal out-of-time-order correlator (OTOC) growing exponentially in time in systems without chaos. The system is a one-dimensional quantum mechanics with a potential whose part is an inverted harmonic oscillator. We numerically observe the exponential growth of the OTOC when the temperature is higher than a certain threshold. The Lyapunov exponent is found to be of the order of the classical Lyapunov exponent generated at the hilltop, and it remains non-vanishing even at high temperature. We adopt various shape of the potential and find these features universal. The study confirms that the exponential growth of the thermal OTOC does not necessarily mean chaos when the potential includes a local maximum. We also provide a bound for the Lyapunov exponent of the thermal OTOC in generic quantum mechanics in one dimension, which is of the same form as the chaos bound obtained by Maldacena, Shenker and Stanford.

Figures (14)

• Figure 1: Inverted harmonic oscillator potential for $\lambda=2, g=1/50$ ($V(0)=12.5$) and its energy levels. The energy levels in red color play an important role for exponential growth of OTOCs. See Fig. \ref{['fig:microcanonical']}. Note that the energy levels smaller than $n=11$ are almost degenerated so the black lines below the top of the hill in (a) are double lines.
• Figure 2: The microcanonical OTOCs for the IHO. We can see the strong exponential growth for intermediate modes (like $n=9\sim 13$), while lower modes and higher modes do not show initial exponential growth. The energy range of these intermediate modes correspond to the local maximum of the potential (the red lines in Fig.\ref{['fig1a']}).
• Figure 3: The time dependence of the thermal OTOCs of the system \ref{['HP']} for various values of the temperature $T$. The dashed part is non-linear and the solid part is linear (exponential in $t$). The time domains for the solid lines are determined at each temperature data such that the linear fit provides the smallest confidence interval of the slope normalized by the slope itself. The obtained time domains are longer than the twice of ${\cal O}(1/\lambda_\text{OTOC})$, which certifies the exponential growth here.
• Figure 4: The temperature dependence of the Lyapunov exponents $\lambda_\text{OTOC}$ of the thermal OTOCs. The bar represents the 95% confidence interval for the exponent. The dashed curve is the fitting function for the Lyapunov exponents obtained in the range $20\leq T \leq 70$.
• Figure 5: Inverted harmonic oscillator potential for $\lambda =2\sqrt{5}, g = 1/10, (V(0) = 62.5)$.