Reversible Quantum Information Spreading in Many-Body Systems near Criticality

Quantum chaotic interacting $N$-particle systems are assumed to show fast and irreversible spreading of quantum information on short (Ehrenfest) time scales $\sim \log N$. Here, we show that, near criticality, certain many-body systems exhibit fast initial scrambling, followed subsequently by oscillatory behavior between reentrant localization and delocalization of information in Hilbert space. We consider both integrable and nonintegrable quantum critical bosonic systems with attractive contact interaction that exhibit locally unstable dynamics in the corresponding many-body phase space of the large-$N$ limit. Semiclassical quantization of the latter accounts for many-body correlations in excellent agreement with simulations. Most notably, it predicts an asymptotically constant local level spacing $\hslash /\tau$, again given by $\tau \sim \log N$. This unique timescale governs the long-time behavior of out-of-time-order correlators that feature quasiperiodic recurrences indicating reversibility.

Figure 1

Phase space portrait of energy $\omega (z,\phi )$, Eq. (7), of the classical dynamics for the three-mode model of attractive bosons in 1D, for different values of interaction $\alpha$. Here, $z$ denotes the relative occupation of the noninteracting ground state and $\phi$ the conjugate angle. For $\alpha >1$, a global energy minimum (cross) and a separatrix (red) connecting two hyperbolic fixed points (dots at $z=1$) appear. Gray lines represent the orbits (tori) that fulfill the quantization condition (10) for $N=20$. Note that the quantized orbits change their character sequentially.

Figure 2

Excitation spectrum $E_m-E_0$ for $N=300$ and $0\le \alpha \le 2$ within three-mode approximation, Eq. (3). Symbols and lines denote numerical and semiclassical results, respectively. The gray dotted line indicates inflection points, the finite size precursor of an excited state QPT, where quantized orbits cross the separatrix (in Fig. 1). Inset: closeup around $\alpha =1$.

Figure 3

(a) Asymptotic $N\gg 1$ density of states $\overline{\rho }(\alpha ,E)$, Eq. (11), showing level bunching (bright straight line) around the precursor of the excited state QPT at $E=E_{\mathrm{sep}}$ in the supercritical regime $\alpha >1$. (b) Slice of $\overline{\rho }(\alpha ,E)$ at $\alpha =2$ compared with numerically calculated inverse level gaps (orange symbols) exhibiting a characteristic logarithmic divergence.

Figure 4

Asymptotic level spacings $E_{[k]}-E_{[k-1]}$ (symbols) obtained from (11) [61], where $E_{[k=0]}$ refers to the level closest to the separatrix energy (excited state QPT), approaching the characteristic constant spacing $\mathrm{\Delta }E$, Eq. (12), (dashed line) for $\alpha =2$ and $N={10}^6,{10}^9,{10}^{13},{10}^{23},{10}^{42}$ (the latter with relevance for black hole physics) from top (blue) to bottom (yellow) within windows with $\sim \log N$ levels. Energies obtained from full semiclassical quantization, (10), (solid line) for $N={10}^6$ confirm the validity of Eq. (11).

Figure 5

Oscillatory time evolution of the OTOC (13) after a quench of the noninteracting condensate across the mean-field critical point, reflecting scrambling and unscrambling (i.e., indicating reversibility) close to criticality for various particle numbers $N$. (a) $C(t)$ exhibits distinct, approximately $\tau$-periodic oscillations where $\tau$, Eq. (12), is the local Ehrenfest time [three-mode model (3) with $\alpha =2$]. Inset: Initial growth of $C(t)$ approaching the exponent $2{\lambda }_{s}t$ (dashed line) with increasing $N$, thereby confirming Eq. (14). (b) Tendency towards periodicity (suggested in gray) for the nonintegrable extended five-mode model ($|k|\le 2$, see [61]) for $\alpha =1.05$, with a period of the form (12) with $\lambda \approx 0.21$, consistently found in the level spacings and the classical instability. The prefactors $C_0$ facilitate the comparison.