Reversible quantum information spreading in many-body systems near criticality

TL;DR

This paper reveals that near criticality, many-body quantum systems exhibit a complex interplay of fast information scrambling and oscillatory localization-delocalization dynamics, with a unique timescale governing reversibility.

Contribution

It demonstrates that quantum critical systems show reversible information dynamics with a characteristic timescale proportional to log N, combining semiclassical analysis and simulations.

Findings

Fast initial scrambling near criticality

Oscillatory localization and delocalization of information

Recurrent behavior indicating reversibility

Abstract

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 $\hbar/\tau$, again given by $\tau\! \sim\! \log N$. This unique timescale governs the long-time behavior of out-of-time-order correlators that feature quasi-periodic recurrences indicating reversibility.

Equations

$$C(t)=⟨[\hat{W}(t),\hat{V}{]}^{†}[\hat{W}(t),\hat{V}]⟩$$

$$\hat{\mathcal{H}}=\int_{0}^{2\pi}\mathop{}\!\mathrm{d}{\theta}\,\hat{\Psi}^{\dagger}(\theta)(-\partial^{2}_{\theta})\hat{\Psi}(\theta)-\frac{\pi\tilde{\alpha}}{2}\big{[}\hat{\Psi}^{\dagger}(\theta)\big{]}^{2}\big{[}\hat{\Psi}(\theta)\big{]}^{2},$$

$$\hat{\Psi }(\theta )=({\hat{a}}_0+{\hat{a}}_{-1}{e}^{-\mathrm{i}\theta }+{\hat{a}}_1{e}^{\mathrm{i}\theta })/\sqrt{2\pi },$$

$$\hat{N}=\sum _k{\hat{a}}_k^{†}{\hat{a}}_k,\hat{K}=\sum _{k}k{\hat{a}}_k^{†}{\hat{a}}_k.$$

$$\tilde{N}\equiv n_{-1}+n_0+n_1,K\equiv n_1-n_{-1}$$

$$\frac{E}{\tilde{N}}=\omega (z,\phi )+\alpha \left(-\frac{1}{4}+\frac{3}{2\tilde{N}}-\frac{9}{8{\tilde{N}}^2}\right)-\frac{1}{\tilde{N}},$$

$$\omega (z,\phi )=(1-z)\left[1-\alpha \left[(1-z)/8+z{\cos }^2\phi \right]\right]$$

$$z=\frac{n_0}{\tilde{N}},\phi ={\vartheta }_0-\frac{1}{2}({\vartheta }_1+{\vartheta }_{-1}),\{z,\phi \}=\frac{1}{\tilde{N}}.$$

$${\lambda }_{\mathrm{s}}^{(1,2)}=2\sqrt{\alpha -1}\equiv {\lambda }_{\mathrm{s}}.$$

$$\frac{1}{2\pi }\oint \mathrm{d}\phi [1-z(\omega ,\phi )]=\frac{m+\frac{1}{2}}{\tilde{N}},$$

$$m=0,1,\dots \lfloor N/2\rfloor ,\tilde{N}=N+3/2,N=0,1,\dots .$$

$$\bar{\varrho }(E)=\frac{-1}{2\pi \lambda }\log \left(\frac{\left|E-E_{\mathrm{sep}}\right|}{\tilde{N}}\right)+\mathcal{O}(1)$$

$$\tau =\frac{2\pi }{\Delta E}=\frac{1}{\lambda }\log \tilde{N}+\mathcal{O}(1)$$

$$C(t)=-N^{-4}⟨\psi |[{\hat{n}}_0(0),{\hat{n}}_0(t){]}^2|\psi ⟩$$

$$C(t)\sim ({\hslash }_{\mathrm{eff}}^2/{\tilde{N}}^2)e^{2{\lambda }_{\mathrm{s}}t},$$

Full text

Quantum phase transition, spreading of correlations, semiclassical methods, scrambling times, many-body quantum systems

The dynamics of quantum information in complex many-body (MB) systems presently attracts a lot of attention Altman (2018); Swingle (2018) ranging from atomic and condensed quantum matter to high energy physics. The evolution of an (excited) quantum MB system towards a state of thermal equilibrium usually goes along with the scrambling of quantum correlations, encoded in the initial state, across the system’s many degrees of freedom. Such dynamics requires an improved understanding of MB quantum chaos and the link with thermalization Srednicki (1994); Rigol et al. (2008); Eisert et al. (2015); Kaufman et al. (2016) and its suppression Nandkishore and Huse (2015); Schreiber et al. (2015); Altman (2018).

Echo protocols, measuring how a perturbation affects successive forward and backward propagations in time, sensitively probe the stability of complex quantum dynamics. Here, out-of-time-order correlators (OTOCs) Larkin and Ovchinnikov (1969); Maldacena et al. (2016); Maldacena and Stanford (2016)

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play a central role, with first experimental implementations Gärttner et al. (2017); Li et al. (2017); Wei et al. (2018), allowing to distinguish various classes of MB systems by their operator growth. On the one side there are slow scramblers, such as systems in the MB localized phase exhibiting logarithmically slow operator spreading Chen (2016); Huang et al. (2017); Fan et al. (2017); Swingle and Chowdhury (2017) or, e.g., Luttinger liquids Dóra and Moessner (2017) showing only quadratic increase. On the other side, an exponentially fast initial growth of OTOCs is commonly viewed as a quantum signature of MB chaos. Examples comprise systems with holographic duals to black holes Maldacena et al. (2016); Cotler et al. (2017), the SYK-model Kitaev (2015); Maldacena and Stanford (2016); Polchinski and Rosenhaus (2016); Patel and Sachdev (2017), and condensed matter systems close to a quantum phase transition (QPT) Shen et al. (2017); Heyl et al. (2018); Alavirad and Lavasani (2019); Chávez-Carlos et al. (2019) or exhibiting chaos in the classical limit of large particle number $N$. In such large-$N$ systems, the exponential growth rate for OTOCs is given by the Lyapunov exponent of their classical counterpart Maldacena et al. (2016); Swingle et al. (2016); Rozenbaum et al. (2017); Bohrdt et al. (2017); Scaffidi and Altman (2017); Rammensee et al. (2018); García-Mata et al. (2018); Chávez-Carlos et al. (2019); Jalabert et al. (2018) and prevails up to the Ehrenfest $\log N$ time where MB quantum interference sets in Rammensee et al. (2018); Tomsovic et al. (2018). Subsequent OTOC time evolution towards an ergodic limit is then often governed by slow classical modes von Keyserlingk et al. (2018).

Here we show that exponentially fast scrambling need not necessarily lead to quantum information loss: There exist systems exhibiting initial growth of complexity without relaxation, i.e., after a quench to an interacting system close to criticality the OTOCs do not show monotonous saturation; instead the correlations imprinted initially can be periodically retrieved.

Quantum critical large-$N$ systems are particularly suited for considering the inter-relation between spreading of correlations, quantified through OTOCs, and corresponding nonlinear classical mean-field (MF) dynamics. There, critical phenomena are often viewed as quantum manifestations of structural changes in classical phase space, associated with unstable MF motion close to separatrices. While corresponding studies Emary and Brandes (2003); Caprio et al. (2008); Bastidas et al. (2014); Stránský et al. (2014); Bastarrachea-Magnani et al. (2016); Rubeni et al. (2017); Pappalardi et al. (2018) commonly invoke a classical MF analysis, we will show that MB semiclassical quantization beyond MF allows for a precise characterization of the locally unstable quantum dynamics. In the language of renormalization group analysis we therefore expect our results to be valid for any dimension within lower and upper critical dimension as long as a MF (classical) limit exists 111Following Chaikin and Lubensky (1995) criticality occurs above the lower, while quantum fluctuations become subdominant above the upper critical dimension..

While in generic quantum chaotic systems the Ehrenfest time $\sim\log N$ is distinctly shorter than the Heisenberg time (associated with the inverse mean level spacing) we will show that these two scales indeed coincide for certain quantum critical systems where an adiabatic separation allows for an effective 1D description. Quantization of their locally hyperbolic MB dynamics implies two inter-related features: Even though the dynamics may be separable, OTOCs still grow exponentially with a rate given by the local MF instability exponent $\lambda_{s}$ up to times $(1/\lambda_{s})\log N$. Second, the inverse mean level spacing in the relevant spectral region also scales as $\log N$. Hence, the quantum critical dynamics is governed by $\log N$ as the sole time scale.

Remarkably, this level spacing turns out to be asymptotically constant, approaching a harmonic oscillator spectrum, although the underlying hyperbolic dynamics is unstable and rather corresponds to an inverted oscillator 222See Molina-Vilaplana and Sierra (2013) for similar results for the $xp$ model in $\mathrm{AdS}_{2}$.. This equidistant level spacing implies strong, periodic quantum recurrences on short $\log N$-scales that dominate OTOCs and hence reflect unscrambling of information in quantum critical MB systems. After showing this behavior in a prototypical integrable model we consider a nonintegrable extension and confirm the robustness of this feature, indicating that it is not linked to integrability but is characteristic for QPTs driven dominantly by a single degree of freedom. On the contrary, in generic chaotic MB systems randomlike evolution is expected for enormously long (Heisenberg) times beyond which the spectral discreteness eventually enforces recurrences 333Such recurrences are exceptional and exist only at particular wavelengths Tomsovic and Lefebvre (1997)..

Quantum critical atomic Bose gas.—As a generic example of critical behavior we consider the 1D attractive Bose gas with periodic boundary conditions (attractive Lieb-Liniger model) Lieb and Liniger (1963); Kanamoto et al. (2003); Sykes et al. (2007). While its Hamiltonian

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with bosonic field operators $\hat{\Psi}$ and $\hat{\Psi}^{\dagger}$, describes quasi-1D ultracold atomic gases with interactions parametrized by $\tilde{\alpha}$ Strecker et al. (2002); Khaykovich et al. (2002); Chin et al. (2010), its MF dynamics is governed by the Gross-Pitaevskii equation. It exhibits a QPT at a critical coupling $\tilde{\alpha}N=1$ Kanamoto et al. (2003, 2005, 2006) where the homogeneous condensate starts forming a bright soliton. Although for finite $N$ eigenvalues of the quantum integrable Hamiltonian (2) can be, in principle, found through Bethe ansatz Sakmann et al. (2005); Sykes et al. (2007), this does not allow for systematically treating the $N\to\infty$ limit, except for special states Flassig et al. (2016); Piroli and Calabrese (2016). Instead we first truncate SMm $\hat{\Psi}(\theta)$ to the lowest three momentum modes,

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as commonly done for exact diagonalization Kanamoto et al. (2005); Dvali et al. (2013); Dvali and Panchenko (2015)—a good approximation for $\tilde{\alpha}N<1$ that also contains all the physics relevant for understanding the QPT and its precursors for $\tilde{\alpha}N\geq 1$ Kanamoto et al. (2003, 2006); Sykes et al. (2007); Dvali et al. (2013). The generalization to the non-integrable 5-mode model will be discussed later. The model (2,3) near the QPT mimics black holes as graviton condensates Dvali et al. (2013); Dvali and Gomez (2014); Dvali and Panchenko (2015), can be essentially realized using ultracold spin-1 atoms Gerving et al. (2012) and has attracted considerable attention Arwas et al. (2015); Prüfer et al. (2018); Garcia-March et al. (2018) for time crystals Kosior et al. (2018).

Classical MF limit.—Besides the energy also particle number and total (angular) momentum are conserved:

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Hence, the truncation to three modes, in contrast to five or more modes Herbst and Ablowitz (1989), renders the system integrable in that its large-$N$ MF limit, formally representing a classical limit, is integrable. This allows for devising a MB version of semiclassical torus quantization Tabor (1989); Ozorio de Almeida (1990) to analytically find the spectrum and wave functions being asymptotically exact for $N\!\to\!\infty$. To this end we write the operators in symmetric order and replace $\hat{a}_{k}\mapsto\sqrt{n_{k}}e^{\mathrm{i}\vartheta_{k}}$ for $k\!=\!-1,0,1$, where $(n_{k},\vartheta_{k})$ are continuous classical conjugate variables. Using

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and considering $K=0$, the classical energy per particle is, defining $\alpha=\tilde{\alpha}\tilde{N}$, SMc

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where the classical dynamics is completely determined by

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with phase space coordinates

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Note that $1/\tilde{N}$ only enters as an effective quantum of action in the Poisson bracket Engl et al. (2014).

The Hamiltonian $\omega(z,\varphi)$ involves different types of classical trajectories following lines of constant energy in phase space with periodicity $\varphi\!\mapsto\!\varphi+\pi$, see Fig. 1. For $\alpha<1$ all trajectories are deformed horizontal lines (rotations). For $\alpha>1$ an island centered around a new minimum energy fixed point emerges with orbits vibrating in $\varphi$, similar as for the pendulum. This goes along with the formation of a separatrix at $E\!=\!E_{\mathrm{sep}}$ ($\omega\!=\!0$) associated with two hyperbolic fixed points at $z\!=\!1$ and characterized by (in)stability exponents

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Semiclassical quantization.—To study genuine quantum effects we go beyond the classical MF picture using semiclassical torus quantization. While related WKB approaches, devised for one-dimensional systems, were successfully used in two-site models Albiez et al. (2005); Gati et al. (2006); Graefe and Korsch (2007), we adapt a multi-dimensional generalization. In the MB context it yields the quantization rules

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Equations (10) effectively quantize the phase space areas bounded by the lines $\omega=\mathrm{const.}$ (shaded areas in Fig. 1 for $m=0$), giving rise to energies $\omega_{m}\!=\!E_{m}/\tilde{N}$, in perfect agreement with results from exact diagonalization, see Fig. 2.

Quantum phase transition.—The QPT for $N\!\to\!\infty$ is associated with the ground state corresponding to the quantized orbit enclosing the phase space area $1/(2\tilde{N})\to 0$ that is always vibrational for $\alpha>1$ if $\tilde{N}$ is large enough. Its energy scales as $\omega_{\mathrm{min}}\sim-(\alpha-1)^{2}$, in contrast to $\omega_{\mathrm{min}}=0$ for $\alpha<1$ where the quantized orbit approaches $z=1$, leading to the nonanalytic dependence on $\alpha$ of the MF ground state at $\alpha\!=\!1$. Precursors of such nonanalyticity for finite $\tilde{N}$ appear for every quantized orbit changing from rotation to vibration upon tuning $\alpha$. This is reflected in the sequence of avoided crossings in Fig. 2 building up an excited state QPT when $\tilde{N}\to\infty$ Caprio et al. (2008); Bastarrachea-Magnani et al. (2016). Remarkably, Eq. (10) even allows us Hummel (2018) to analytically obtain the scaling laws 444We are not aware of renormalization group calculations of critical exponents in this model.

governing the approach of $\Delta\alpha_{\mathrm{gap}}\sim N^{-2/3}$ and $E_{\mathrm{gap}}\sim N^{-1/3}$ (see inset of Fig. 2) to their MF values, in perfect agreement with numerical and heuristic observations Kanamoto et al. (2003, 2005); Dvali and Panchenko (2015).

Hence, MB semiclassical quantization goes beyond the Bogoliubov picture of Kanamoto et al. (2003, 2006) where the excitation spectrum collapses to zero at the MF critical coupling $\alpha\!=\!1$. Instead, for finite $\tilde{N}$ we find huge accumulations of levels around the separatrix. This precursor of an excited state QPT leads to characteristic features in the spectra and to the emergence of a local log time scale. An asymptotic $N\gg 1$ analysis SMs of (10) that generally holds close to a separatrix SMu yields the average density of states

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with a characteristic logarithmic divergence at $E=E_{\rm sep}$ (see Fig. 3). Here $\lambda\!=\!\lambda_{\mathrm{s}}/2\!=\!\sqrt{\alpha-1}$, Eq. (9), and the $\mathcal{O}(1)$ finite size correction involves a system specific timescale related to the traversal along the separatrix SMu ; SMs .

Asymptotically constant level spacing and log time.—Most notably, evaluating (10) close to $E_{\rm sep}$ one finds SMe that a set of levels, growing in number logarithmically with $N$, becomes asymptotically equidistant with level spacing $\Delta E$, see Fig. 4. The associated time scale SMl

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is the Heisenberg time corresponding to the local spectral gap $\Delta E$, but exhibits a striking similarity with the Ehrenfest time $\tau_{\rm E}\!=\!(1/\lambda_{\rm L})\log\hbar_{\rm eff}^{-1}$ with $\hbar_{\rm eff}\!=\hbar$ and $1/\tilde{N}$ in chaotic single-particle Berman and Zaslavsky (1978), respectively, MB systems Rammensee et al. (2018) with Lyapunov exponent $\lambda_{\rm L}$. This justifies to relate (12) to a local Ehrenfest time associated with the dynamical instability characteristic of critical behavior. Due to the universality of (11) and (12), supported by the classical renormalization group analysis of Zaslavsky (2005), this turns out to be a generic behavior close to hyperbolic fixed points. The crossover of the Heisenberg time from usually algebraic in $N$ to log time behavior is not shared by generic chaotic systems.

Out-of-time-order correlator.—We address the drastic consequences of this transition for the evolution of quantum MB correlations: We quantify the spreading of information through the OTOC , Eq. (1),

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for operators $\hat{n}_{0}(t)/N$. In chaotic systems quasi-classical arguments Larkin and Ovchinnikov (1969); Maldacena et al. (2016); Swingle et al. (2016) confirmed by MB semiclassical theory Rammensee et al. (2018) predict a short-time behavior $C(t)\sim\hbar^{2}e^{2\lambda_{\mathrm{L}}t}$ passing into a saturation regime at $\tau_{E}$. Although the system (3) is integrable, we can use similar arguments to predict that in the quantum critical regime the dynamical instability close to the hyperbolic fixed points also causes such an exponential behavior,

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but with a rate given by the (in)stability exponents $\lambda_{\mathrm{s}}$, Eq. (9) (see Pappalardi et al. (2018) for a related result for chains of large spins).

Within the present MB model we have numerical access to huge particle numbers and hence can thoroughly check the commonly assumed exponential growth of OTOCs in the truly semiclassical large-$N$ limit, as well as associated log-time effects. In Fig. 5 we present numerical results for $C(t)$ computed from (13) after imposing an interaction quench to the non-interacting ground state $|\psi\rangle=1/\sqrt{N!}(\hat{a}_{0}^{\dagger})^{N}|0\rangle$. The inset displays the short time behavior of $\log C(t)$ up to the Ehrenfest time scale $\tau_{E}=(1/\lambda_{\mathrm{s}})\log\tilde{N}$ for $N\!=\!10^{2}$ to $10^{6}$, showing convergence to the slope $2\lambda_{\mathrm{s}}t$ (dashed line) predicted in (14).

Unscrambling of correlations.—Figure Fig. 5(a) shows that instead of monotonously approaching a plateau for $t>\tau_{\mathrm{E}}$, as in generic chaotic MB systems, $C(t)$ exhibits distinct oscillations with a period given by the log time $\tau$, (12), that includes finite-size corrections SMl and hence slightly differs from $\tau_{E}$. Since $C(t)$ is a measure of information spreading, these oscillations reflect reversibility of quantum information flux in Hilbert space as a result of genuine MB interference. This is supplemented by corresponding oscillations in the evolution of entanglement encoded in the one-body entropy SMl . Close to criticality, $C(t)$ is dominated by an increasing (with $\log N$) number of states close to $E_{\rm sep}$ SMl where the spectrum gets asymptotically equidistant (Fig. 4). This induces revivals (getting more and more pronounced with increasing $N$) associated with a unique time scale, the log time $\tau$, (12), taking the role of a Heisenberg time close to criticality. To clearcut show this asymptotic periodicity, here deduced from MB separatrix quantization, requires large-$N$ regimes that, to the best of our knowledge, are not accessible with present numerical methods for chaotic MB systems.

To assess the latter and to explore the generality of our findings we further relax the truncation (3) to five modes ($|k|\leq 2$), implying unstable non-integrable dynamics. Within the bounds of numerical tractability ($N=\mathcal{O}(10^{4})$) we verify that this quantum critical MB system, despite being non-integrable, exhibits the same tendency towards periodic revivals, see Fig. 5(b). Via adiabatic separation we can again attribute this behavior to locally hyperbolic MF dynamics of a single dominant degree of freedom SMf . Moreover, we find the same interrelation between the MF instability $\lambda_{\mathrm{s}}$, the period $\tau$, and the local level spacing as in the integrable three-mode model.

In conclusion, by means of many-body semiclassical quantization we could explain the exponentially fast scrambling and buildup of correlations in quantum systems that are critical, but not globally chaotic. We uncovered a generic mechanism for fast dynamics near a quantum critical point. Moreover, for large $N$ we demonstrated the emergence of nearly equidistant spectra giving rise to recurrences in OTOCs on $\log N$-time scales, resembling features of time crystals Kosior et al. (2018). Their observation should be in experimental reach since, e.g., recurrences based on stable dynamics have already been observed in a system with thousands of atoms Rauer et al. (2018). Our analysis of the non-integrable 5-mode model shows that such memory effects do not require integrability and indicates their existence in larger classes of critical systems with a dynamical decoupling of a dominant unstable mode from other collective degrees of freedom. Moreover, our results shed light on generic mechanisms governing the dynamics at (excited-state) quantum phase transitions beyond mean-field.

We acknowledge funding through the Studienstiftung des Deutschen Volkes (BG) and the Deutsche Forschungsgemeinschaft through project Ri681/14-1.