Universal level statistics of the out-of-time-ordered operator

TL;DR

By introducing the Lyapunovian operator, the paper links semiclassical Lyapunov growth with universal spectral fluctuations in a chaotic quantum system. Using the quantum stadium billiard, the authors compute the spectra of log-OTOC operators and find GOE/GUE statistics in the bulk depending on operator type and filtering, and they present a phase-space picture that interpolates between independent finite-time exponents and quantum interference. They further demonstrate a time-dependent transition in projected Lyapunovians from Poisson to Wigner-Dyson statistics as phase-space correlations develop, connected to Ehrenfest time via a Moyal-bracket framework. The work provides a versatile tool for diagnosing quantum chaos and clarifying quantum-to-classical correspondence beyond traditional OTOC measures.

Abstract

The out-of-time-ordered correlator has been proposed as an indicator of chaos in quantum systems due to its simple interpretation in the semiclassical limit. In particular, its rate of possible exponential growth at $\hbar \to 0$ is closely related to the classical Lyapunov exponent. Here we explore how this approach to quantum chaos relates to the random-matrix theoretical description. To do so, we introduce and study the level statistics of the logarithm of the out-of-time-ordered operator, $\hatΛ(t) = \ln \left( - \left[\hat{x}(t),\hat{p}_x(0) \right]^2 \right)/(2t)$, that we dub the "Lyapunovian" or "Lyapunov operator" for brevity. The Lyapunovian's level statistics is calculated explicitly for the quantum stadium billiard. It is shown that in the bulk of the filtered spectrum, this statistics perfectly aligns with the Wigner-Dyson distribution. One of the advantages of looking at the spectral statistics of this operator is that it has a well-defined semiclassical limit where it reduces to the matrix of uncorrelated classical finite-time Lyapunov exponents in a partitioned phase space. We provide a heuristic picture interpolating these two limits using Moyal quantum mechanics. Our results show that the Lyapunov operator may serve as a useful tool to characterize quantum chaos and in particular quantum-to-classical correspondence in chaotic systems, by connecting the semiclassical Lyapunov growth at early times, when the quantum effects are weak, to universal level repulsion that hinges on strong quantum interference effects.

Figures (5)

• Figure 1: Energy-level statistics for quantum stadium billiard (separate for each eigenstate parity, combined ParityComment). Contribution from the bouncing-ball modes Graf92Sieber93Alt99 is removed within the spectrum unfolding. Solid line shows GOE Wigner-Dyson distribution.
• Figure 2: Eigenvalue-spacing distribution for the bulk of the Lyapunovian spectrum for every second state (within each parity block, combined). The total number of levels is $10^5$. Insets: (a) bulk level spacing distribution for $\hat{\Gamma}(t=0)$; (b) the same for $\hat{\Gamma}(t\neq0)$. Solid lines show the corresponding Wigner-Dyson distributions.
• Figure 3: Eigenvalue-spacing distribution for the bulk of the spectra of an ensemble of projections of $\hat{C}^{(1)}(t)$ onto the coherent-state subspaces averaged over that ensemble and over time in two ranges of time: (a) at $t<t_E$, the distribution shows clear signatures of the Poisson component related to the uncorrelated nature of the phase space; (b) at $t>t_E$, the statistics tends to the universal GUE Wigner-Dyson distribution as phase-space correlations build up. With larger matrices, one can see that it becomes exact, such as the one shown in Fig. 2(b) in the main text. The low quality of the histograms is related to the small size of the subspaces ($8\times8$ matrices).
• Figure 4: Schematics of the correlation development in phase space with time if initial states are semiclassical. (a) At times $t \ll t_E$, the local finite-time Lyapunov exponents are independent in different cells. (b) As time goes towards $t_E$, the correlations build up. (c) Around $t_E$, the phase-space becomes fully correlated, as shown by the distributions in Fig. \ref{['fig:GUE_GSE']}.
• Figure 5: OTOC as the operator (\ref{['L']}) averaged over the initial state (\ref{['eq:InitialState']}) at early times (semi-log scale). $\hbar_{\rm eff}=2^{-7},\;x_0=y_0=0,\;p_{0x}/p_{0y}=e,\;\sigma=1/\sqrt{2}$. Between $t_c$ and $t_E$, the growth is nearly exponential, $C(t)\propto e^{2\tilde{\lambda} t}$, for time longer than $4/(2\tilde{\lambda})$, but the value of $\tilde{\lambda}$ is not self-averaged yet.