Gauging classical and quantum integrability through out-of-time-ordered correlators

Out-of-time-ordered correlators (OTOCs) have been proposed as a probe of chaos in quantum mechanics, on the basis of their short-time exponential growth found in some particular setups. However, it has been seen that this behavior is not universal. Therefore, we query other quantum chaos manifestations arising from the OTOCs, and we thus study their long-time behavior in systems of completely different nature: quantum maps, which are the simplest chaotic one-body system, and spin chains, which are many-body systems without a classical limit. It is shown that studying the long-time regime of the OTOCs it is possible to detect and gauge the transition between integrability and chaos, and we benchmark the transition with other indicators of quantum chaos based on the spectra and the eigenstates of the systems considered. For systems with a classical analog, we show that the proposed OTOC indicators have a very high accuracy that allow us to detect subtle features along the integrability-to-chaos transition.

Figure 1

Sketch of the typical time dependence of the OTOC for one-body and finite-size many-body systems exhibiting different behavior in the short- and long-time regimes.

Figure 2

Main panel: (a) example of the typical behavior of $C\left(t\right)$ across a transition from integrability to chaos. The system is the Harper map described in Sec. 4c2, having $K=0.063$ (large amplitude oscillations, violet), $0.19$ (medium amplitude oscillations, green), and 0.75 (small amplitude oscillations, blue), with $D=200$. Insets: (b) Short-time of the OTOC for the previous values of $K$, but with $D=1024$. The same color code is used, and the symbols are open circles, squares, and circles, by increasing values of $K$. The red line shows $\sim e^{2{\lambda }_{\mathrm{L}}t}$; (c–e) Phase portraits for $K=0.063,0.19,0.75$ (from left to right) using the same color code as before.

Figure 3

Main panel: (a) typical behavior of $C_{zz}\left(t\right)/{〈C_{zz}〉}_t(l=1)$ across the transition from integrability to chaos for the Ising model with tilted magnetic field described in Sec. 4d2. The chain length is $L=8 D=256$. The chosen angles are $\theta =0.08\pi /2$ (large amplitude oscillations, violet), $0.31\pi /2$ (medium amplitude oscillations, green), and $0.79\pi /2$ (small amplitude oscillations, blue). Upper panels: (b-d) normalized OTOC FFT distribution of frequencies ${\left|{\tilde{C}}_{zz}\left(\omega \right)\right|}^2$, from the data of the main panel (a) for increasing values of $\theta$ (from left to right) using the same color code as before.

Figure 4

(a) ${\overline{\xi }}_E$ (filled circles) and the $\beta$ parameter (open circles) resulting from the eigenvalues and eigenvectors of the Standard map with $D=1000$. The red solid line is $r_{\mathrm{ch}}$ (top) obtained from the corresponding classical map, as described in Appendix pp2 with $n_{\mathrm{tot}}=35000$ and averaging the curves obtained for $t_{max}=490,491,...,509$. In the inset we show the values of ${\overline{\rho }}^2$ (black) obtained from the Berry-Robnik distribution and $\eta$ obtained from the ratios distribution. (b) Normalized ${\overline{\xi }}_{{}_{\mathrm{OTOC}}}$ (filled circles) and ${\left({\overline{\sigma }}_{{}_{\mathrm{OTOC}}}\right)}^{-1}$ (open circles) for $D=600$. The number of iterations is $6\times {10}^3$. The red solid line corresponds to $1/r_{\mathrm{reg}}=1/(1-r_{\mathrm{ch}})$, normalized as described in the text.

Figure 5

Same as Fig. 4 for the Harper map with $D=1000$ (a, b) and $D=200$ (c). The number of iterations is $2\times {10}^4$. The red solid lines were obtained using $n_{\mathrm{tot}}=35000$ and averaging the curves obtained for $t_{\max }=90,91,...,109$.

Figure 6

(a) Blowup of a section of Fig. 5 for values of $K$ around 1.5. Bottom panels: The phase portraits for the classical Harper map with $K=1.44$ (b), 1.47 (c), 1.508 (d), 1.571 (e), and 1.696 (f). The lines indicate the position of the corresponding $K$ values on the axis in the top panel. Points inside regular islands are drawn darker to enhance contrast.

Figure 7

Chaos transition for the perturbed XXZ spin chain. (a) ${\overline{\xi }}_{{}_E}$ (filled circles), Brody parameter $\beta$ (open circles) for the even parity subspace in a spin chain of length $L=13$ and $N=5(D^{\mathrm{Even}}=651)$. Inset (a): The values of ${\overline{\rho }}^2$ (black) and $\eta$ (orange) for the previous system conditions. (b) ${\overline{\xi }}_{{}_{\mathrm{OTOC}}}$ (filled circles) and ${\overline{\sigma }}_{{}_{\mathrm{OTOC}}}^{-1}$ (open circles) for a spin chain of length $L=13$ and $N=5$ (the entire Hilbert space has $D=1287$) and $l=1$. Inset (b): ${\overline{\sigma }}_{{}_{\mathrm{OTOC}}}^{-1}$ for a spin chain of length $L=10$ and $N=6$ (the full Hilbert space has $D=210$) for $l=1\text{(triangles)},2\text{(diamonds)},3\text{(circles),}$ and $4\text{(asterisks)}$.

Figure 8

Short-time growth of $C_{zz}\left(l,t\right)$ for the perturbed XXZ model with $L=9, N=5$, different spin separations $l=1$ (a), 2 (b), 3 (c) (solid lines), and different values of $\lambda =0,0.5,1$. Short-time power-law growth predicted with the HBC formula of Eq. (A11) is also shown (dashed lines).

Figure 9

Chaos transition for the Ising model with tilted magnetic field. (a) ${\overline{\xi }}_{{}_E}$ (filled circles) and Brody parameter $\beta$ (open circles) for the even parity subspace for a spin chain length $L=12(D^{\mathrm{Even}}=2080)$. (b) ${\overline{\xi }}_{{}_{\mathrm{OTOC}}}$ (filled circles) and ${\overline{\sigma }}_{{}_{\mathrm{OTOC}}}^{-1}$ (open circles) for a spin chain of length $L=8(D=256)$. It is important to remark that the plot begins at $\theta =0.03\pi /2$.

Figure 10

Short-time growth of scaled $C_{zz}\left(l,t\right)$ for an Ising chain of length $L=8$ subjected to a tilted magnetic field with different angles $\theta$ and various spin separations sites $l=1$ (a), 2 (b), 3 (c) (solid lines). The short-time power-law growth predicted by Eq. (A10) is also plotted (dashed lines).

Figure 11

Chaos transition for the Heisenberg spin chain with random magnetic field. (a) ${\overline{\xi }}_{{}_E}$ (filled circles) and Brody parameter $\beta$ (open circles) in a chain of length $L=13, N=5(D=1287)$ and averaged over 100 realizations. (b) ${\overline{\xi }}_{{}_{\mathrm{OTOC}}}$ (filled circles) and ${\overline{\sigma }}_{{}_{\mathrm{OTOC}}}^{-1}$ (open circles) for a chain of length $L=9$ and $N=5(D=126)$ and averaged over 100 realizations. The plot begins at $h=0.05$.

Figure 12

Short-time growth of $C_{zz}\left(l,t\right)$ for the Heisenberg spin chain with random magnetic field and a chain of length $L=9, N=5$ for different spin separations $l=1,2,3$ (solid lines). Short-time power-law growth is predicted with the HBC formula (A7) (dashed lines).

Figure 13

(a, c) classical phase portrait for the standard map (a) with $K=3.0$ and the Harper map (c) with $K=0.5$. (b, d) evolution in time for $30000$ randomly chosen initial conditions and the color represents $t_{\max }$. Light colors stand for small evolution times, where the point has returned to the vicinity of the initial position before the $t_{\max }$. The darkest color stands for the cases where the point has not returned to the neighborhood of the initial condition within a time $t_{\max }$. The ratio (B1) is $r_{\mathrm{ch}}=0.882$ for the standard map (b) and $r_{\mathrm{ch}}=0.866$ for the Harper map (d), meaning that $86\%$–$88\%$ (approx.) of the area of the phase space is chaotic.