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

Out-of-time-order 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 set-ups. 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 classical analogue, we show that the proposed OTOC indicators have a very high accuracy that allow to detect subtle features along the integrability-to-chaos transition.


I. INTRODUCTION
The original Bohr-Sommerfeld formulation of Quantum Mechanics addressed integrable classical systems, with as many conserved quantities as degrees of freedom.Einstein's 1917 observation that such a quantization scheme remained extremely limited (as integrability is a singularity among dynamical systems) remained relatively unnoticed until the late fifties, probably due to the success of Schrödinger equation [1,2].The quantization of chaotic systems, as well as the understanding of the consequences of classical chaos on quantum observables such as the level statistics, developed in the seventies and eighties [3,4], provided the connection of Quantum Mechanics with fully chaotic systems, which constitute another singularity within the ensemble of dynamical systems.The connection of classical and quantum properties in the generic case of mixed systems, away from the two previously mentioned singularities, remains, comparatively, less understood and more difficult to quantify.
Two important aspects spur our interest for the intermediate behavior between fully integrable and completely chaotic regimes.On the one hand, in quantum systems without classical analog the notion of integrability is still valid, although defined through the separability and the soluble character of the quantum problem.On the other hand, in multidimensional and many-body systems the "no-man's land" between the two singular behaviors is difficult to avoid.
Within this context, the recent impressive development of experimental techniques for many-body quantum systems [5][6][7][8][9], monitoring in time complex processes like localization or thermalization, enhanced the need to understand quantum dynamics and its connection with the concepts of integrability, chaos and ergodicity.A useful tool towards this task, which has lately received considerable attention is the out-of-time ordered correlator (OTOC), defined from the commutator of two operators V and Ŵ (t) (the Heisenberg time evolution of operator Ŵ ) as where the angular brackets denote the average over an initial state.While this time-dependent quantity was first considered in a semi-classical study of superconductivity [10], the present interest results from its use as a measure of quantum information scrambling [11][12][13][14][15][16][17][18][19], which in addition, is accessible to experiments [9,[20][21][22].
For chaotic many-body systems the scrambling measured by the OTOC was conjectured to increase exponentially in time, with a temperature-dependent bound on the growth-rate [11].The strongly-coupled, exactlysolvable Sachdev-Ye-Kitaev many-body quantum model [23,24], saturates the bound, while being is dually related to black holes via the AdS/CFT correspondence [25].
The exponential short-time behavior has been demonstrated to hold in some many-body systems such as the Dicke [26] and the Sachdev-Ye-Kitaev [27,28] models, and the corresponding systems have then been dubbed as "fast scramblers" [29].Systems with a classical counterpart, like quantum maps and billiards, also exhibit an exponential growth of the OTOC, with a rate that can be either equal [30,31] or proportional [32][33][34] to the Lyapunov exponent, depending on the initial state.However, the exponential growth does not constitute a universal behavior.Other examples, like spin-chains in the presence of random fields [13], Luttinger liquids [35], or weakly chaotic systems [36] exhibit a polynomial increase of the OTOC (and have then been dubbed as "slow scramblers").
While the exponential growth of the OTOC for relatively short times was the initial focus for the above-cited studies, it latter appeared that the long-time properties of the OTOC were equally interesting from a Quantum Chaos point of view [30,37].
The comparatively fewer studies of the long-time behavior of the OTOC [30,31,33,[37][38][39] have been centered on the saturation obtained in finite-size classically chaotic systems.A semi-classical theory for fully chaotic many-body systems linked the saturation with quantum interference [37].In the simpler case of quantum maps, the approach to the saturation was shown to be exponential, and dominated by the largest non-trivial Ruelle-Pollicott resonance of their chaotic classical counterparts [30].More recently the long-time regime in the case of two interacting maps was also considered [40,41].In addition, the appearance of oscillations in the long-time regime for critical many-body systems was studied in [42].The long-time behavior of the OTOC is particularly interesting in view of its consequences for thermalization processes in many-body systems [5,[43][44][45][46].
The goal of this work is to try to provide a connection between the degree of integrability and the characteristic features of the OTOC dynamics for the case of mixed systems.We will show that such a connection is firmly established using the long-time dynamics of the OTOC, where we can match quantum and classical chaos indicator with proposed OTOC indicators.
In order to test the universality of the established connection, we study one-particle systems having a classical counterpart (two quantum maps) where a parameter can be used to tune the transition from integrability to chaos, as well as many-particle systems without a classical analog (three different spin chains) where chaos is typically driven by an interaction parameter and characterized by the nearest-level spacing-distribution.
This work has the following structure.In Sec.II we review the OTOC and its most relevant properties while in Sec.III two of the canonical quantum chaos indicators are exposed.Both previous sections provide the mechanisms to analyze the systems studied in this work.In Sec.IV we present the physical systems to be used in numerical simulations, consisting of Quantum Maps (Sec.IV C) and three many body Spin chain systems (Sec.IV D).We present our conclusions and outlook in Section V.In the App.A the mathematical details of the calculations for the short-time behavior in spin chains are included and in App.B we explain a method to measure the area of the chaotic region in phase space.

II. SHORT AND LONG-TIME BEHAVIOR OF THE OUT-OF-TIME-ORDERED CORRELATOR
We will work with initial thermal states in the infinite temperature limit, for which Ô = T r Ô/D , where D is the dimension of the Hilbert space.Moreover, taking the Ŵ and V operators to be Hermitian, the OTOC defined in Eq. (1) becomes Typically, the time-dependence of the OTOC appears as schematically presented in Fig. 1, with two welldefined time regimes.The short-time growth of the

Short time
Long time OTOC is given by the operator spread, or scrambling, where the initial quantum information spreads over the available degrees of freedom in a quantum system.As discussed in the introduction, the short-time growth is exponential in many cases [11,26,30,32,34,37].But such a behavior of fast scramblers is not generic.Examples of slow scramblers have been predicted for weakly chaotic systems, were the short-time growth has been shown to be linear [36], for Luttinger liquids, where a quadratic initial-state-independent behavior has been obtained [35], for a random-field XX spin-chain, where the initial growth exhibits a power-law given twice the distance between the sites associated with the chosen operators [13], as well as for the anomalous phases of the interacting Aubry-André model [47].The rich, non-universal initial behavior of the OTOC hinted on the usefulness of using it in order to characterize different many-body phases [15][16][17][18][19][47][48][49].

Short time growth
The dotted region defined in violet in the sketch of Fig. 1 stands for the variety of possible outcomes for the OTOC's short-time growth.Even if in this work we concentrate ourselves in the infinite-temperature limit, it is worth to mention that in the general case of an initial thermal state, the extent of the growth regime can be strongly dependent on the temperature.For instance, the exponentially increasing regime in the case of a chaotic billiard appears in a limited time-window that shrinks as the temperature increases [31].While most of the results concerning the initial growth of the OTOC were obtained through numerical calculations, there exist some analytic results.Among them, the short-time exponential growthrate for the "cat map" that has been analytically shown to be given by the classical Lyapunov exponent [30], and the semi-classical approaches that allowed to establish a connection with the classical Lyapunov exponent for a chaotic stadium billiard [31] and for an interacting boson system [37].
As stated in the introduction, the long-time behavior of the OTOC in the chaotic case is signed by a clear sat-uration.[30,31,33,[37][38][39].Unitarity, ergodicity and finite-size yield, for the definition (2) of the infinite temperature case, a limiting value of 1.In the general case of an initial thermal state, the limiting value depends on the chosen operators, and for the canonical choice of position and momentum operators, it is proportional to the temperature [31,38].
Classically integrable systems do not show a clear saturation of the OTOC, and the long-time limit is signed by strong oscillations.In the case of the square billiard, large periodic oscillations arising from the commensurability character of its energy spectrum prevent the approach to saturation in the long-time limit [38].The typical long-time behavior for the intermediate case of mixed systems is sketched in Fig. 1, where irregular oscillations are superposed to a saturation value.The main result of this paper is the characterization of these aperiodic oscillations in very different systems, and linking this information with the one stemming from other quantum and classical chaos indicators.

III. QUANTUM CHAOS INDICATORS
We will characterize different mixed quantum systems with two widely used Quantum Chaos indicators in order to gauge the transition from integrability to chaos.The first indicator is the Brody parameter β, obtained by fitting the level-spacing distribution P (s) with the Brody distribution [50], defined as where Γ (x) is the gamma function.The Brody distribution P B (s) approaches to a Poisson distribution P P (s) for β → 0 and resembles the Wigner-Dyson (WD) distribution P W D (s) when β → 1.Since the seminal works by Berry and Tabor [51] and Bohigas, Gianoni, and Schmidt [4] it is by now well established that a Poisson distribution is associated to non-ergodic, regular systems while a behavior resembling a Gaussian ensemble and characterized by a WD distribution is to be associated to quantum chaotic dynamics.
The second indicator that we consider is the localization of eigenstates, characterized by the inverse participation ratio (IPR).Suppose |ψ i is an eigenstate of the system of interest written in an arbitrary basis {|φ j } N −1 j=0 as |ψ i = a ij |φ j .We will denote the IPR of the eigenstate as the inverse of the second moment of the distribution elements Therefore ξ E is a measure of localizaton relative to the original basis and is defined as where small values of ξ E characterize a localized eigenstate while larger values signal delocalization.For systems with a WD distribution the coefficients a ij are independent random variables.These type of states are completely delocalized, having the direct consequence that ξ deloc E ≈ N/3, because the coefficients |a ij | 2 fluctuate [52,53].In the numerical calulations we will consider the average over all the eigenstates We note that this measure is basis dependent so it should be analysed with care.

A. Overview
We start by presenting numerical studies of the OTOC in the long-time regime using a one-body system (the Harper map, whose precise description is given in Sec.IV C 2) in which the transition from integrability to chaos is clear from the available classical counterpart.In Fig. 2 such a transition is obtained by varying the parameter K of the corresponding Hamiltonian, and represented by the color change from violet to green, and then to blue.The panels (c), (d), and (e) present the corresponding classical phase portraits as a Poincare surface of section for K = 0.063, 0.19, and 0.75, respectively.The panels (a) and (e) show, respectively, the long and short-time regimes of the OTOC C(t) of the quantum map for the three different cases.For short times the growth of C(t) is strongly dependent on K, and only in the completely chaotic case an exponential growth with the Lyapunov exponent (red line) can be identified, as it was shown in [30].For the very long times of the main panel C(t) oscillates around the saturation value.In the integrable case a strong oscillatory behavior is observed (violet curve).The oscillations are characterized by a large amplitude and a seemingly small number of frequency components.The amplitude of the oscillations decreases as the chaos parameter K becomes larger (green curve), approaching small quasi-random fluctuations around a constant saturation value in the fully chaotic regime (blue curve).
The presented behavior is generic to other one-particle systems (data not shown).
The conclusions extracted from the previously discussed one-body example carry over to the many-body case.Fig. 3 panel (d) shows the long-time behavior of the OTOC for an Ising chain with a tilted magnetic field (described in SubSec.IV D 2).The transition from integrability to chaos is driven by the angle θ and represented by the color change from violet to green, and then to blue.Like in the one-body example, the oscillations around the saturation value decrease upon approaching the chaotic limit.The insets portray the magnitude square of the Fourier transform C (ω), which helps to characterize the long-time OTOC oscillations and is used to define the ξ OT OC parameter of Eq. ( 6).

B. Long-time indicators for the OTOC
The generic behavior of the OTOC presented in the previous section lead us to conjecture that measuring and quantifying the oscillatory behavior of C(t) in the long-time regime allow to assess the chaotic nature of the quantum system.The suggested link between the integrability of a quantum system and the long-time oscillations of C(t) makes it necessary to develop quantum indicators that are able to gauge the importance of these oscillations.
A direct quantity to be used in order to characterize the oscillations is the standard deviation 2 , where the averages are taken over intermediate time-windows.A second useful quantity is the localization in Fourier space, obtained by computing the inverse participation ratio of the Fourier transform To avoid the initial transient and a resulting large peak around ω = 0, we compute the Fourier transform of across the transition from integrability to chaos for the case of an Ising chain with a tilted magnetic field are given in the insets of Fig. 3. Just like the previously defined ξ E , a small ξ OT OC characterizes a very localized signal, meaning a small number of frequencies are present, which is characteristic in the weakly chaotic chase.On the contrary a large ξ OT OC corresponds to delocalization in frequency space and an almost constant value for C(t).In the following sections we test if the long-time regime of the OTOC can detect the quantum chaos transition in paradigmatic models of one and many body system.
We note that we will consider systems that depend on a parameter and we are interested in the variation of ξ OT OC and σ OT OC with the parameter.To compare both quantities behavior in a unique plot, we normalize them with respect to their maximum value present in the studied parameter ranges, σ−1 C. One-body systems: Quantum maps Classical maps on the 2-torus are the simplest systems which can have all the essential features of chaotic motion.Here we will consider quantum maps on the torus which are the quantized counterparts of a classical canonical transformation corresponding to these classical maps [54,55].The torus structure implies periodicity in position and momentum.This periodicity results upon quantization in a discrete Hilbert space of dimension N , and an effective Planck constant h eff = 1/(2πN ).Given a classical map M the corresponding quantum map U M is then a unitary operator with an N × N matrix representation, and the time evolution is given in discrete steps by U t M , with t an integer.Quantum maps have been extensively used to study quantum chaos [56] and irreversibility [57].Moreover there exist efficient quantum algorithms for many of the well known quantum maps [58][59][60][61], making them interesting test beds of quantum chaos in experiments using quantum simulators.
The OTOC that we will consider for maps is where X(t) = ( Û † ) t X Û t , and are Hermitian operators defined in terms of the unitary Schwinger shift operators [62].If |q , |p are position and momentum states, related by p |p = e −2πiqp/N (with q, p = 0, . . ., N − 1) , then In the semiclassical limit X and P approximate position and momentum operators.
The two maps which we consider, described below, are derived from kicked systems so they can be written as The advantage is that numerical implementation of evolution (and diagonalization [63]) is very efficient using fast Fourier transformations.For both of the maps that we consider, the kicking strength is the chaos parameter.

Standard map
The quantum (Chirikov) standard map (SM) [56] U corresponds to the classical map For small values of K the dynamics are regular.Below a certain critical value K c the motion in momentum is limited by KAM curves.These are invariant curves with irrational frequency ratio (or winding number) which represent quasi-periodic motion, and they are the most robust orbits under nonlinear perturbations [64].At K c = 0.971635 . . .[65], the last KAM curve, with most irrational winding number, breaks.Above K c there is unbounded diffusion in p. K, there might exist islands but the motion is essentially chaotic.
In Fig. 4 we show the numerical results obtained for for the standard map.In the top panel the Brody parameter β and ξE exhibit the same transition from localized (nonergodic) behavior to delocalized (ergodic) behavior.The red curve is a Metropolis sampling approximation r ch of the area of the chaotic region for the classical map, described in App.B. The direct correlation between these quantities is evident.
In the bottom panel we show ξOTOC and (σ OT OC ) −1 .We use the relative variance (dividing σ OT OC by the time average of C(t)) because this average changes with K and also because it reflects a relative deviation from the mean.We can also see a transition from localized to delocalized but not taking place quite at the same values.The remarkable observation is that that these two quantities seem to reproduce the behavior of r−1 reg ≡ r reg /r min reg (also defined in in App.B).This is the inverse normalized value of r reg = 1 − r ch which measures the area in phase space of the integrable region.

Harper map
The quantum Harper map (HM) is an approximation of the motion of kicked charge in the presence of an external magnetic field [66].The corresponding classical map is We will only consider the case K 1 = K 2 = K.For K < 0.11, the classical dynamics is regular, while for K > 0.63 it is fully chaotic for most values of K, although there are some particular values where small islands appear [67].In Fig. 5 (top, panel) we see that for the Harper map β does not change much with K.This is due to the fact that the map has symmetries, and it can be solved using an irrational h = 2π = 1/N (see Refs. [63,68]).
For historical reasons we only consider a rational h, and therefore an approximately constant β is obtained as expected.On the other hand a transition can be observed for the ξE .As shown for the standard map, the classical chaotic area r ch follows well the behavior of ξ E (red curve) In the bottom panel we see that the same qualitative behavior can be observed for ξOTOC and (σ OT OC ) −1 .Similar to what happens with the standard map, a transition is visible, taking place at a value somewhat different from the one suggested by ξ E .However, as noted above, ξOTOC and σOTOC follow surprisingly well the behavior of rreg , which implies that we can relate their behavior to the size of the regular islands in the corresponding classical phase space.
It is known that for the classical Harper map, upon increasing K, there are values of for which there appear regular islands and then disappear.Such an effect translates into dips of r reg .It is interesting to see, as it is shown in detail in Fig. 6, that the K-dependences of ξOTOC and σOTOC reproduce the shape of the dip very accurately.Thus, the proposed OTOC indicators seem to be well-suited to identify chaotic regions and also a mixed phase space regime.

D. Many-body systems: Spin chains
We consider three many-body spin-1/2 systems described by a generic Hamiltonian that depends on a tunable parameter (e.g.interaction strength).By changing this parameter the system is driven through a transition from integrable to chaotic regimes.From a quantum chaos point of view, the transition can be observed through the spectra of eigenvalues or in eigenstate distributions.These type of systems have been extensively used in studies of quantum thermodynamics [69][70][71][72] and many-body localization [73][74][75].
In the following text is set to 1, while L refers to the number of spin-1/2 sites in the chain and are the spin operators at site i = 0, 1, ..., L−1, with σµ the corresponding Pauli matrix associated with the direction µ = x, y, z.Boundary conditions for all the spin chain models are set as open.Since the spin operators are both unitary and Hermitian, the OTOC can be written for infinity temperature as where D is the dimension of the Hilbert space.
1. Perturbed XXZ model The first model we consider is a spin-1/2 chain with nearest-neighbour (NN) spin interactions and a perturbation consisting in next-nearest-neighbour (NNN) interactions tuned by a strength parameter λ.The hamiltonian of the chain is with This system has been extensively studied from the Quantum Chaos point of view in Ref. [76].It presents a chaotic regime when the NNN coupling strength λ becomes comparable with the NN coupling, turning the level-spacing distribution from Poisson to WD.The latter transition occurs when all symmetries are removed.For this reason, in our calculations, the parameter µ is fixed at 0.5 to avoid conservation of total spin Ŝ2 , which occurs at µ = 1.The total spin of the system in the z direction Ŝz =

L−1 i=0
Ŝz i is also conserved.This symmetry allows separation of the total spanned space, of dimension D, into smaller subspaces ŜN with a fixed number N of spins up or down.The dimension of each subspace ŜN is The system also presents a symmetry with the parity operator Π, defined with the permutation operators as where the defined operator Π is described for a spin chain of odd length L (the even L situation is analogous).The conservation of Π divides the spanned space into odd and even subspaces with dimensions D = D Even + D Odd , where D Even/Odd ≈ D/2.
Similarly to the case presented in the previous section of quantum maps, we now study how the effect of the integrability to chaos transition, occurring in the eigenstates and eigenvalues spectral translates into the long-time properties of the OTOC.
In the top panel of Fig. 7 we show such a transition in the quantum chaos indicators β and ξE as a function of the NNN coupling strength λ.In this calculations we analyzed a chain of length L = 13 and N = 5, where the even parity subspace has D Even = 651 states.Although it is more pronounced for β than for ξE , we clearly see in both measures that a transition occurs between λ = 0.3 and λ = 0.5.The inverse participation ratio is computed with respect to the spin site basis and averaged over 10% of the values in the center of the energy spectrum.
In the bottom panel of Fig. 7 we present the measures σOTOC and ξOTOC of the long-time behavior of the OTOC C zz (l, t).In this case, the separation of the spin operators sites is l = 1 and similar results were obtained for other separation values l.In the inset of Fig. 7 we show σ−1 OT OC as a function of the parameter λ for l = 1, 2, 3 and 4, for a small spin chain (L = 10 and N = 6).We observe that the same qualitative behavior as in β and ξE , i.e. a transition as a function of λ, is observed for σ−1 OT OC , showing that it is a good indicator of an integrable to chaotic transition.We remark that in the case of the OTOC such a transition is already revealed for much smaller chain lengths than the ones needed to observe it for β.
We end the analysis of the perturbed XXZ spin chain considering the short time growth of the OTOC.In Fig 8 we show such a regime for different spin separations l = 1, 2, 3 and perturbation strength λ.We can clearly see that the behaviour is characterized by a power-law Eq. (A11) that was obtained in the Appendix using the HBC formula.As is evident from Eq. (A11) and the data shown in Fig 8, the short-time power law growth is strongly dependent on the coupling strength λ.We remark that this short time regime is not influenced by the integrable to chaotic transitions shown in the top panel of Fig 7.

Ising model with tilted magnetic field
The second model consists of a spin-1/2 chain with NN interactions (Ising model) with a tilted magnetic field.The Hamiltonian of the system is Parameters are set at J = 2 and B = 1.When the angle θ = 0, the magnetic field is longitudinal and when θ = π/2, it becomes the transverse field model.In both cases the system is integrable with a highly degenerate spectrum.The Jordan-Wigner (JW) [77] transformation yields the solution of a non-interacting fermionic system.At intermediate angles 0 < θ < π/2, the model undergoes a quantum chaos transition.In this case, JW transformation maps the system to a model of interacting fermions.The quantum chaos transition and eigenvalues spectral properties have been studied in [78], in our analysis a WD NN level spacing distribution occurs near θ = π/4.The parity symmetry described in the previous model is also present in this system and therefore, even and odd subspaces will be analyzed separately for the eigenstate/eigenvalue spectral properties.The OTOC analysis will focus on the entire Hilbert space of dimension D = 2 L .It is important to notice that at θ = 0 the OTOC C zz (l, t) = 0, and as θ increases, the OTOC continually increases it's maximum value.Therefore, because our interest arises in the OTOC oscillations with respect to it's mean value, both ξOTOC and σ−1 OT OC are analyzed for C zz (l, t) / C zz (l) t , where C zz (l) t is the temporal average after the OTOC reaches it's mean saturation value.
In Fig. 9 we analyze the quantum chaos transition in the spectral properties and in the long-time regime of the OTOC.In the top panel we show β and ξE as a function of the tilt angle θ for a spin chain of length L = 12.The calculations were done with the even subspace which has D Even = 2080 states.As we have previously noted, the system is integrable for θ = 0 and π/2, where β(0) takes negative values since the NN distribution resembles more a Delta distribution than a Poisson one and β(π/2) ≈ 0. The WD distribution is reached at θ ≈ π/4 where β = 1.In the bottom panel of Fig. 9 we show ξOTOC and σ−1 OT OC for a L = 8 spin chain length.The choice such that the lower panel uses a very small chain is intentional to highlight an interesting property.Both results not only show nearly identical behaviour, but the choice of a very small spanned Hilbert space dimension also highlights the fact that the OTOC does not require Hilbert spaces as large as those required by the statistical studies of eigenstate/eigenvalue properties.We checked that qualitatively equivalent results for larger spin chains can be obtained (data not shown).
In Fig 10 we show the short time behavior of the OTOC C zz (l, t) for separation sites l = 1, 2 and 3 and angles θ = 1/4π, 3/8π and π/2 of the magnetic field.The angle θ = π/2 has been thoroughly studied in Ref. [79], and the short time power-law formula presented in that work gets barely modified by the presence of angle θ.The relation obtained by the HBC formula Eq. (A10) is also plotted (see Appendix A) and clearly describes the shot-time regime.As shown in Fig 10, although angle dependence is present in Eq. (A10), it does not affect the short-time power-law.Furthermore, like in the previous model, the transition to chaos does not affect the short time growth in any way.

Heisenberg spin chain with random magnetic field
The last model that we analyze consists of a spin chain with NN interactions (the Hamiltonian of Eq. 18 with µ = 1) coupled with an external random magnetic field in the z-direction [80].The Hamiltonian of the system is where h i are independent random variables at each site, uniformly distributed between [−h, h].This is a paradigmatic prototype model that has been used to study the many-body localization (MBL) transition.[81][82][83][84] The transition in the level spacing statistics has been a subject of study for quite some time [84][85][86][87].For h = 0 and taking into account symmetries it can be shown that the system is solvable and the nearest level spacing distribution is Poissonian.As h increases the disorder breaks the symmetries and the system starts to become chaotic reaching a Wigner-Dyson distribution at h ≈ 0.5, corresponding to ergodicity.Finally if the disorder is too strong there is a MBL transition (for a review see [88]).As in the perturbed XXZ chain, total spin in the z direction is conserved and therefore subspaces ŜN with a fixed number N of spins up or down are used for the calculations.To study the Eigenstate IPR the diagonalized basis components are compared with the spin site basis and averaged over 10% of the values in the center of the energy spectrum, obtaining ξE .Because of the statistical nature of the random variables h 1 , ..., h n , ξE is also averaged over several realizations but no new notation is added to prevent confusion.We remark that it is not the scope of this work to try to identify the MBL transition present for large values of h [88].However we note that the decline of ξOTOC and σ−1 OT OC with increasing h points in the right direction to identify the MBL behavior.
Finally, we show in Fig. 12 the short time growth of the OTOC C zz for this spin chain.We also plot the short time power-law relation that was obtained in the Appendix A. We can see that Eq.A7 describes very well this time regime.The random parameters h 1 , ...h N , which cause the system to transition into chaotic regimes and then to the MBL, do not play a role in the short time growth power-law.

V. CONCLUSIONS
The OTOC is a quantity that has drawn attention recently because it has been suggested as a measure of quantum chaos but also because of possible implications in studies of high energy physics, many-body localization and quantum information scrambling and thermalization.
After the so-called scrambling time the OTOC establishes around a constant mean value.However there are fluctuations that remain and we had evidence that fluctuations where strongly correlated to the dynamical properties of the system.We have proposed two quantities to assess these long-time fluctuations, the spectral IPR and the time variance, which we compared with well established chaos indicators like the Brody parameter and the localization of the eigenstates.We computed these quantities numerically for various one-body and many-body systems, which where known to undergo a transition to chaos depending on one parameter.
From our simulations we conclude that the fluctuations of the OTOC can be used to characterize a transition to chaos in quantum systems.For systems with a classical counterpart the spectral IPR can be directly related to the area of the regular islands in phase space.For manybody spin systems the same qualitative behavior has been observed even though there is no classical counterpart.Besides, we also show that all the considered spin chains are slow scramblers due to the power-law growth of the OTOC for short time that does not depend on the regular to chaos transition.
Our results indicate that the main features of the dynamics can be extracted from the long time of the OTOC.They suggest that this regime is feature rich and deserves more attention and study, in particular because the long time regime matters in problems of current interest like quantum thermalization.They can also have implications in light of recent experimental advances.

Figure 1 .
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 .
Figure 2. Main panel: (a) example of the typical behavior of C(t) across a transition from integrability to chaos.The system is the Harper map described in SubSec.IV C 2 having K = 0.063 (violet) 0.19 (green), and 0.75 (blue), with N = 200.Insets: (b) Short-time of the OTOC for the Harper map for the previous values of K, but with N = 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 ∼ e 2λ L t ; (c,d,e) Phase portraits for the three values of K using the same color code, K = 0.063 (c) 0.19 (d), and 0.75 (e).

Figure 4 .
Figure 4. Top panel: ξE (filled circles) and the β parameter (open circles) resulting from the eigenvalues and eigenvectors of the Standard map with N = 1000.Bottom panel: (filled circles) Normalized ξOTOC (open circles) (σ OT OC ) −1 for N = 600.The number of iterations is 6 × 10 3 .The red solid line is r ch (top) and 1/rreg obtained from the corresponding classical map, as described in App.B with ntot = 25000 and tmax = 250.

Figure 5 .K 1 K
Figure 5. as Fig. 4 for the Harper map with N = 1000 (top panel), and N = 200 (bottom panel).The number of iterations is 2 × 10 4 .The red solid lines were obtained using ntot = 20000 and tmax = 200 e 2 s D f 7 o m R w X p f Z P W d y y / y c c l c Z M Z e a d k t p z s 6 j N y M e 0 Y W 2 L 3 d R x V d U W F J s 3 K m q B b Y l n i + O c a 2 B W T D 2 g T H M / K 2 b n V F N m / f e E i Y I f r J S S q t w l E 7 D N k K T u E k c k 0 V S N / V b 4 8 X o Y z D R t g 4 l o Q z 7 e L B n Y g u P S W 3 I o E q 4 U 6 I h E P X c v e i 3 q 3 f d / a o D l I f o H j e u P X K I l P m g a f 1 6 y e M x l c N L r k u 1 u 7 3 s c 7 X 1 F 8 1 p D 7 9 E H 9 A k R t I P 2 0 D d 0 h A a I o T H 6 i X 6 h 6 + A w E I E J 6 r k 1 6 N x l t t C D C q 5 u A e e m 1 B E = < / l a t e x i t > 1.5 < l a t e x i t s h a 1 _ b a s e 6 4 = " 3 S U D G p O N C F I + E V 6 3 4 3 d e + c k z y + o = " > A A A C x H i c d V H d S h w x F M 6 O t t q x P 2 o v e x M c C l 4 t k y 3 S 3 h S l l d J L y 7 o q 7 A x L J n N m D Z t k h i T T s s T x D b z V B + h l H 6 S P U P o 2 z c 4 q 6 K 4 9 E P j 4 f j j n 5 G S V 4 M b G 8 d 9 O s L L 6 5 O n a + r N w 4 / m L l 6 8 2 t 7 Z P T F l r B g NW i l K f Z d S A 4 A o G l l s B Z 5 U G K j M B p 9 n k 8 0 w / / Q 7 a 8 F I d 2 2 k F q a R j x Q v O q P V U n 3 T 3 R p t R 3 I 3 b w s u A 3 I J o / 3 f 4 s f r 5 J z w a b X V + J X n J a g n K M k G N G Z K 4 s q m j 2 n I m o A m T 2 k B F 2 Y S O Y e i h o h J M 6 t p Z G / z W M z k u S u2 f s r h l 7 y c c l c Z M Z e a d k t p z s 6 j N y M e 0 Y W 2 L D 6 n j q q o t K D Z v V N Q C 2 x L P F s c 5 1 8 C s m H p A m e Z + V s z O q a b M + u 8 J E w U / W C k l V b l L J m C b I U n d B Y 5 I o q k a + 6 3 w 4 / U w m G n a B h P R h n y 8 W T K w B c e F t + R Q J F w p 0 B G J e u 5 O 9 F r U u + v / v w G W h + g f N q 4 / c o m W + L B p / H n J 4 j G X w U m v S 9 5 1 e 9 / i 6 O A T m t c 6 e o N 2 0 C 4 i 6 D 0 6 Q F / R E R o g h s b o C l 2 j m + B L I A I T 1 H N r 0 L n N v E Y P K r j 8 B + x i 1 B M = < / l a t e x i t > 1.7< l a t e x i t s h a 1 _ b a s e 6 4 = " n 1 w T N a u x q h a h B S T j b Q i h / q R L H p 8 = " > A A A C x H i c d V H N T t w w E P a G 0 t K U l p 8 e e 7 E a V e p p F W 8P c E F F L U I 9 U m 0 X k D b R y n E m i 7 W 2 E 9 k O a G X C G 3 C F B + D Y B + k j V H 2 b e r M g l V 0 6 k q V P 3 4 9 m x p N V g h s b x 3 8 6 w c q z 1 e c v 1 l 6 G r 9 Z f v 9 n Y 3 N o + N m W t G Q x Y K U p 9 m l E D g i s Y W G 4 F n F Y a q M w E n G S Tr z P 9 5 B y 0 4 a X 6 Y a c V p J K O F S 8 4 o 9 Z T f d L d G W 1 G c T d u C y 8 D c g + i z 7 / C v e r u d 3 g 0 2 u r 8 T P K S 1 R K U Z Y I a M y R x Z V N H t e V M Q B M m t Y G K s g k d w 9 B D R S W Y 1 L W z N v i D Z 3 J c l N o / Z X H L / p t w V B o z l Z l 3 S m r P z K I 2 I 5 / S h r U t d l P H V V V b U G z e q K g F t i W e L Y 5 z r o F Z M f W A M s 3 9 r J i d U U 2 Z 9 d 8 T J g o u W C k l V b l L J m C b I U n d J Y 5 I o q k a + 6 3 w 0 / U 4 m G n a B h P R h n y 8 W T K w B c e l t + R Q J F w p 0 B G J e u 5 B 9 F r U e + j / v w G W h + g f N K 4 / c o m W + K B p / H n J 4 j G X w X G v S z 5 1 e 9 / j a P 8 L m t c a e o f e o 4 + I o B 2 0 j 7 6 h I z R A D I 3 R N b p B t 8 F h I A I T 1 H N r 0 L n P v E W P K r j 6 C / E e 1 B U = < / l a t e x i t > 1 < l a t e x i t s h a 1 _ b a s e 6 4 = " R f y 0 7 a Y M / e C b H R a n 9 U x a 3 7 L 8 J R 6 U x U 5 l 5 p 6 T 2 3 C x r c / I x b V j b Y i 9 1 X F W 1 B c U W j Y p a Y F v i + d o 4 5 x q Y F V M P K N P c z 4 r Z O d W U W f 8 5 Y a L g J y u l p C p 3 y Q R s M y S p m + G I J J q q s d 8 K P 1 4 P g 5 m m b T A R b c j H m x U D W 3 L M v C W H I u F K g Y 5 I 1 H P 3 o t e i 3 n 3 / / w 2 w O s R x v 3 H H I 5 d o i f t N 4 8 9 L l o + 5 C k 5 6 X f K x 2 z u K o 4 P P a F F r 6 B 1 6 j 3 Y Q Q b v o A H 1 F h 2 i A G A J 0 i a 7 Q r 6 A f f A 8 u A r O w B p 2 7 z D Z 6 U M H s L 7 X r 0 5 w = < / l a t e x i t > 0.5 < l a t e x i t s h a 1 _ b a s e 6 4 = " R c 5 / / K F d n I q + x Y / 9 f 3 4 T X m y C 5 6 w r B r b i O P e W A s u U K 4 U 6 S q K B u x a 9 F g 2 u + / 9 v g P U h h o e t G 0 5 c q i U 9 b F t / 3 m T 1 m O v g e N B P X v U H n + L o 4 B 1 Z 1 h Z 5 R p 6 T l y Q h r 8 k B + U i O y I g w M i W X 5 B v 5 H n w I R G C C Z m k N e l e Z p + R G B R d / A e o C 1 B I = < / l a t e x i t > ⇠OTOC < l a t e x i t s h a 1 _ b a s e 6 4 = " u a B D Q 2 S D / o r o i + V c 7 o 5 p d 6 o r s k Y = " > A A A C 1 n i c d V H N T h s x E H a 2 P 9 D 0 L 5 R j L x a r S j 1 F u + m B H l H h 0 B t U E I g U r 1 Z e 7 2 y w Y n t X t r c 0 M s s N c e U N u F L 1 i f o e f Y A 6 G 5 B K Q k e y 9 O n 7 0 Y x n s k p w Y 6 P o d y d 4 8 v T Z 8 7 X 1 F 9 2 X r 1 6 / e d v b e H d s y l o H v t e e P l Y 6 6 C 4 0 E / / t Q f f P N 3 / o I W t Y 7 e o y 3 0 E c V o G + 2 g r + g A D R F D M 3 S D b t H P Y B R c B J f B 1 c I a d O 4 y m + h B B d d / A c G L 3 F 4 = < / l a t e x i t > (¯ OT OC ) 1 < l a t e x i t s h a 1 _ b a s e 6 4 = " 7 / w X g x G F I a z G g 8 z N M F E 7 e 8 c B H U 0 = " > A A A C 4 H i c d V H N b h M x E H a W v x J + m s I R I V l d I Z U D 0 W 4 4 w L G i P X B r U Z u 2 U r y s Z h 1 v a s X 2 r m w v K H J 9 4 g A 3 x J U n A H G D p + E 9 e A C c T S v R p I x k 6 d P 3 o 5 n x F L X g x i b J 7 0 5 0 7 f q N m 7 f W b n f v 3 L 1 3 f 7 2 3 8 e D I V I 2 m b E g r U e m T A g w T X L G h 5 V a w k 1 o z k I V g x 8 V 0 Z 6 4 f v 2 P a 8 E o d 2 l n N M g k T x U t O w Q Y q 7 z 3 e I g V o R w y f S P C 5 y 9 3 e 4 d 6 O 9 0 / f u m e p z 3 t x 0 k / a w q s g P Q f x 9 m Z Z f v w z / L a f b 3 R + k H F F G 8 m U p Q K M G a V J b T M H 2 n I q m O + S x r A a 6 B Q m b B S g A s l M 5 t o 9 P H 4 S m D E u K x 2 e s r h l / 0 0 4 k M b M Z B G c E u y p W d b m 5 F X a q L H l y 8 x x V T e W K b p o V D Y C 2 w r P P w W P u W b

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0 3 E a D 9 y F G L R 4 c N H / f w O s D n G w 6 9 1 B 7 o i W e N f P z 5 s u H 3 M V H A 3 6 6 f P + 4 E 2 4 8 y u 0 q D X 0 C G 2 i L Z S i F 2 g b v U b 7 a I g o + o C + o 5 / o V 1 R E n 6 L P 0 Z e F N e q c Z x 6 i S x V 9 / Q s r x N / 3 < / l a t e x i t > 0 < l a t e x i t s h a 1 _ b a s e 6 4 = "G p k u 9 2 R 1 o M N M z h E P 4 o Y M Y n N A M l g = " > A A A C w n i c d V H N b t N A E N 6 Y v 2 L + W j h y W W E h c Y r s c C i X i g p y 4 N i q p K 0 U W 9 F 4 P U 6 X 7 K 7 d 3 T U o 2 v g J e i 1 P w J E H 4 R E Q b 8 P G a S W a l J F W + v T 9 a G Z2 8 l p w Y + P 4 T y + 4 c / f e / Q d b D 8 N H j 5 8 8 f b a 9 8 / z Y V I 1 m O G K V q P R p D g Y F V z i y 3 A o 8 r T W C z A W e 5 L O P S / 3 k K 2 r D K / X Z z m v M J E w V L z k D 6 6 n D e L I d x f 2 4 K 7 o J k i s Q v f 8 V 7 t U / f o c H k 5 3 e z 7 S o W C N R W S b A m H E S 1 z Z z o C 1 n A t s w b Q z W w G Y w x b G H C i S a z H W T t v S 1 Z w p a V t o / Z W n H / p t w I I 2 Z y 9 w 7 J d g z s 6 4 t y d u 0 c W P L d 5 n j qm 4 s K r Z q V D a C 2 o o u 1 6 Y F 1 8 i s m H s A T H M / K 2 V n o I F Z / z l h q v A b q 6 Q E V b h 0 h r Y d J5 l b 0 C h J N a i p 3 4 r e X j e D u Y Y u m I o u 5 O P t h o G t O R b e U m C Z c q V Q R 0 k 0 c N e i 1 6 L B d f / / D b A 5 x N G w d U c T l 2 p J h 2 3 r z 5 u s H 3 M T H A / 6 y d v + 4 D C O 9 j + Q V W 2 R l + Q V e U M S s k v 2 y S d y Q E a E E S Q X 5 J J 8 D 4 b B l + A 8 M C t r 0 L v K v C A 3 K l j 8 B b O N 0 5 s = < / l a t e x i t > l a t e x i t s h a 1 _ b a s e 6 4 = " R f y

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0 7 a Y M / e C b H R a n 9 U x a 3 7 L 8 J R 6 U x U 5 l 5 p 6 T 2 3 C x r c / I x b V j b Y i 9 1 X F W 1 B c U W j Y p a Y F v i + d o 4 5 x q Y F V M P K N P c z 4 r Z O d W U W f 8 5 Y a L g J y u l p C p 3 y Q R s M y S p m + G I J J q q s d 8 K P 1 4 P g 5 m m b T A R b c j H m x U D W 3 L M v C W H I u F K g Y 5 I 1 H P 3 o t e i 3 n 3 / / w 2 w O s R x v 3 H H I 5 d o i f t N 4 8 9 L l o + 5 C k 5 6 X f K x 2 z u K o 4 P P a F F r 6 B 1 6 j 3 Y Q Q b v o A H 1 F h 2 i A G A J 0 i a 7 Q r 6 A f f A 8 u A r O w B p 2 7 z D Z 6 U M H s L 7 X r 0 5 w = < / l a t e x i t > l a t e x i t s h a 1 _ b a s e 6 4 = " R f y 0 7 a Y M / e C b H R a n 9 U x a 3 7 L 8 J R 6 U x U 5 l 5 p 6 T 2 3 C x r c / I x b V j b Y i 9 1 X F W 1 B c U W j Y p a Y F v i + d o 4 5 x q Y F V M P K N P c z 4 r Z O d W U W f 8 5 Y a L g J y u l p C p 3 y Q R s M y S p m + G I J J q q s d 8 K P 1 4 P g 5 m m b TA R b c j H m x U D W 3 L M v C W H I u F K g Y 5 I 1 H P 3 o t e i 3 n 3 / / w 2 w O s R x v 3 H H I 5 d o i f t N 4 8 9 L l o + 5 C k 5 6 X f K x 2 z u K o 4 P P a F F r 6 B 1 6 j 3 Y Q Q b v o A H 1 F h 2 i A G A J 0 i a 7 Q r 6 A f f A 8 u A r O w B p 2 7 z D Z 6 U M H s L 7 X r 0 5 w = < / l a t e x i t > 0 < l a t e x i t s h a 1 _ b a s e 6 4 = " G p k u 9 2 R 1 o M N M z h E P 4 o Y M Y n N A M l g = " > A A A C w n i c d V H N b t N A E N 6 Y v 2 L + W j h y W W E h c Y r s c C i X i g p y 4 N i q p K 0 U W 9 F 4 P U 6 X 7 K 7 d 3 T U o 2 v g J e i 1 P w J E H 4 R E Q b 8 P G a S W a l J F W + v T 9 a G Z 2 8 l p w Y + P 4 T y + 4 c / f e / Q d b D 8 N H j 5 8 8 f b a 9 8 / z Y V I 1 m O G K V q P R p D g Y F V z i y 3 A o 8 r T W C z A W e 5 L O P S / 3 k K 2 r D K / X Z z m v M J E w V L z k D 6 6 n D e L I d x f 2 4 K 7 o J k i s Q v f 8 V 7 t U / f o c H k 5 3 e z 7 S o W C N R W S b A m H E S 1 z Z z o C 1 n A t s w b Q z W w G Y w x b G H C i S a z H W T t v S 1 Z w p a V t o / Z W nH / p t w I I 2 Z y 9 w 7 J d g z s 6 4 t y d u 0 c W P L d 5 n j q m 4 s K r Z q V D a C 2 o o u 1 6 Y F 1 8 i s m H s A T H M / K 2 V n o I F Z / z l h q v A b q 6 Q E V b h 0 h r Y d J 5 l b 0 C h J N a i p 3 4 r e X j e D u Y Y u m I o u 5 O P t h o G t O R b e U m C Z c q V Q R 0 k 0 c N e i 1 6 L B d f / / D b A 5 x N G w d U c T l 2 p J h 2 3 r z 5 u s H 3 M T H A / 6 y d v + 4 D C O 9j + Q V W 2 R l + Q V e U M S s k v 2 y S d y Q E a E E S Q X 5 J J 8 D 4 b B l + A 8 M C t r 0 L v K v C A 3 K l j 8 B b O N 0 5 s = < / l a t e x i t > q < l a t e x i t s h a 1 _ b a s e 6 4 = " X E 0 i A j 5 r K e D l z J R R O z S r N g o k P J 4 = " > A A A C w n i c d V H N b h M x E H a W v z b 8 t X A E I Y s V E q d o N z 3 A s Y I c O D Y q a S t l V 9 G s d z Y 1s b 1 b 2 w u K n D 3 2 1 C s 8 B S / Q V + E Z e I k 6 m 1 a i S R n J 0 q f v R z P j y S r B j Y 2 i P 5 3 g 3 v 0 H D x 9 t b X c f P 3 n 6 7 P n O 7 o s j U 9 a a 4 Y i V o t Q n G R g U X O H I c i v w p N I I M h N 4 n M 0 + L / X j 7 6 g N L 9 V X O 6 8 w l T B V v O A M r K e G Z 5 O d M O p F b d F N E F + D c P / 1 5 f D v + Z v L g 8 l u 5 3 e S l 6 y W q C w T Y M w 4 j i q b O t C W M 4 F N N 6 k N V s B m M M W x h w o k m t S 1 k z b 0 n W d y W p T a P 2 V p y / 6 b c C C N m c v M O y X Y U 7 O u L c m 7 t H F t i 4 + p 4 6 q q L S q 2 a l T U g t q S L t e m O d f I r J h 7 A E x z P y t l p 6 C B W f 8 5 3

Figure 8 .
Figure 8. Short time growth of Czz (l, t) for the Perturbed XXZ model with L = 9, N = 5, different spin separations l = 1, 2, 3 (solid lines) and different values of λ = 0, 0.5, 1.Short time power-law growth is predicted with the HBC formula of Eq. (A11) is also shown in dashed line.

1 OTOCFigure 9 .
Figure 9. Chaos transitions for the Ising model with tilted magnetic field.Top panel: ξE (filled circles) and Brody parameter β (open circles) for the even parity subspace for a spin chain length L = 12 (D Even = 2080).Bottom panel: ξOTOC (filled circles) and σ−1 OT OC (open circles) for a spin chain of length L = 8 (D = 256).It is important to remark that the plot begins at θ = 0.03π/2.

Figure 10 .
Figure 10.Short time growth of Czz (l, t) for the Ising model with tilted magnetic field of length L = 8 and different spin separations sites l = 1, 2, 3. Short time power-law growth predicted by Eq. (A10) is also plotted.

1 OTOCFigure 11 .
Figure 11.Chaos transitions for the Heisenberg spin chain with random magnetic field.Top panel: ξE (filled circles) and Brody parameter β (open circles) in a spin chain of length L = 13, N = 5 (D = 1287) and averaged over 100 realizations.Bottom panel: ξOTOC (filled circles) and σ−1 OT OC (open circles) for a spin chain of length L = 9 and N = 5 (D = 126) and averaged over 20 realizations.The plot begins at h = 0.05.

1 Figure 12 .
Figure12.Short time growth of Czz (l, t) for the Heisenberg spin chain with random magnetic field and a spin chain of length L = 9, N = 5 and different spin separations l = 1, 2, 3. Short time power-law growth is predicted with the HBC formula.