On systems of maximal quantum chaos

A remarkable feature of chaos in many-body quantum systems is the existence of a bound on the quantum Lyapunov exponent. An important question is to understand what is special about maximally chaotic systems which saturate this bound. Here we provide further evidence for the `hydrodynamic' origin of chaos in such systems, and discuss hallmarks of maximally chaotic systems. We first provide evidence that a hydrodynamic effective field theory of chaos we previously proposed should be understood as a theory of maximally chaotic systems. We then emphasize and make explicit a signature of maximal chaos which was only implicit in prior literature, namely the suppression of exponential growth in commutator squares of generic few-body operators. We provide a general argument for this suppression within our chaos effective field theory, and illustrate it using SYK models and holographic systems. We speculate that this suppression indicates that the nature of operator scrambling in maximally chaotic systems is fundamentally different to scrambling in non-maximally chaotic systems. We also discuss a simplest scenario for the existence of a maximally chaotic regime at sufficiently large distances even for non-maximally chaotic systems.

One of the key dynamical concepts characterizing a quantum many-body system is scrambling of quantum information. Suppose we slightly disturb a system by inserting a local fewbody operator. Under time evolution, the operator will grow in physical and internal spaces (if there is a large number of single-site degrees of freedom), during which quantum information of the original disturbance is scrambled among more and more degrees of freedom.
While for a general many-body system such evolutions are extremely complicated, it has been increasingly recognized during the last decade that there are remarkable universalities in this process [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. For example, out-of-time-ordered correlators (OTOCs) have emerged as important probes of scrambling in chaotic systems. In particular, in chaotic systems with a large number of single-site degrees of freedom, the commutator squares of generic few-body operators grow exponentially with time at early times (below N characterizes the number of single-site degrees of freedom) with a quantum Lyapunov exponent λ bounded 1 by [5] λ ≤ λ max = 2π β 0 . (1. 2) The bound, which has no classical analogue, follows from unitarity and analyticity.
A variety of systems which saturate the bound have also been found, including holographic systems, SYK and its variations, two-dimensional CFTs with a large central charge (and a gap), CFT correlation functions in the light-cone limit, and so on [1, 2, 8-10, 14, 16-20].
What are special about these maximally chaotic systems? One common theme that has emerged in such systems is that finite temperature dynamics of stress tensor appears to underlie the observed maximally chaotic behavior. In other words, the chaotic behavior appears to be "hydrodynamic" in nature.
It is tempting to use the hydrodynamic nature of chaos as a defining feature of maximally chaotic systems. In this paper we would like to present further support for this idea, and to discuss a few general properties of maximally chaotic systems.
In [21], we proposed an effective field theory for describing chaotic behavior based on "quantum" hydrodynamics. While the theory was motived by maximally chaotic systems, at the time it was not completely clear whether it works only for maximally chaotic systems.
In this paper we first present a simple argument which shows that a system described by 1 For notational consistency with [21], we denote the inverse temperature by β 0 . Henceforth we set = 1.
such an EFT must be maximally chaotic. We thus find strong support, if not a proof, that for chaos to be hydrodynamic in nature requires the system to be maximally chaotic. It also follows that various implications of the EFT discussed in [21] should be considered as distinctive features of maximally chaotic systems. For instance pole-skipping in the energydensity two-point function, first discussed in [21,22], has been argued to be a 'smoking-gun' signature of the hydrodynamic nature of chaos and has since been shown to occur in a widevariety of maximally chaotic systems [19,20,[23][24][25][26][27]. See also [28] for a nice discussion of how pole-skipping generalises to non-maximally chaotic systems.
In this paper we will highlight another implication of the EFT: the exponential growth in the commutator square (1.1) is suppressed at leading order for maximally chaotic systems.
We also illustrate the suppression of the commutator square in SYK models and holographic theories 2 . In particular we give a general argument showing that in holographic theories the commutator square is determined at leading order in 1/N by stringy corrections 3 .
It has been observed in various examples that even non-maximally chaotic systems could exhibit maximally chaotic behavior at sufficiently large distances [4,16,29,30]. For such systems, the chaos EFT can apply at large distances. We discuss a simplest scenario for such a phenomenon.
The plan of the paper is as follows. In Sec. II we give a quick review of key features of the chaos EFT. In Sec. III we show that the EFT describes systems with maximal chaos.
In Sec. IV we discuss suppression of the commutator square (1.1). In Sec. V we discuss a simplest scenario for the existence of a maximally chaotic regime at sufficiently large distances for non-maximally chaotic systems. Sec. VI concludes with a brief discussion of future questions. Appendix A contains some technical details for Sec. IV. 2 The suppression of exponential growth in commutator squares in these models is implicit in the results presented in [4,14,29,31]. 3 The importance of stringy corrections in determining the form of the commutator square in SYK models is discussed in [29].
In this section we present a quick review of the chaos EFT introduced in [21], which is based on the following elements (to avoid cluttering notations below we first use 0 + 1 dimensional systems as an illustration): 1. One imagines the scrambling of a generic few-body operator V (t) at a finite temperature allows a coarse-grained description in terms of building up a "hydrodynamic cloud," i.e. we can write (2.1) HereV (t) is a "bare" operator involving the original degrees of freedom of V , and σ(t) is a chaos mode which describes the growth of the operator in the space of degrees of freedom. Bare operators which involve different degrees of freedom have no correlations, i.e. VŴ = 0 for generic V = W .
2. One identifies the chaos mode σ with the collective degrees of freedom associated with energy conservation, and thus its dynamics is governed by hydrodynamics. However, in order to capture the growth of V (t), one needs a "quantum hydrodynamics" which is valid in the regime ∆t ∼ β 0 , with ∆t typical time scales of interests. This quantum hydrodynamic theory can be considered as a generalization of the conventional hydrodynamics without doing a derivative expansion 4 , and is non-local. Its action, which also incorporates dissipative effects, can be written using the techniques developed in [32,33]. The explicit form of the action S EFT [σ] is not important for our discussion below (we refer interested readers to [21] for details) except that the Lagrangian depends on σ only through derivatives and the equilibrium solution is given by 3. To describe a chaotic system, we still need to impose an additional shift symmetry e −λσ → e −λσ + c tion of σ appears to be self-consistent for any value of λ. In particular, stress-tensor correlation functions do not contain an exponentially growing piece, and fluctuationdissipation relations of the stress tensor do not put any constraint on the value of λ.
We will, however, show in Sec. III that when coupled to a generic operator through (2.1), where L (1) is some differential operator acting on bothV and . Invariance under the shift symmetry implies the condition where subscript t in L (1) t refers to the variable of the derivatives and g V (t) is the two-point function ofV It was also found in [21] that in order for the exponentially growing behavior to cancel for all possible configurations of TOC it is also necessary to impose the version of (2.7) with λ → −λ, i.e.

L
(1) The shift symmetry may be considered as an emergent "macroscopic" symmetry. 5 It has many implications: (i) constrains the structure of correlation functions to all derivative orders; (ii) implies existence of an exponentially growing mode the exchange of which leads to exponential growth of OTOCs; (iii) leads to a connection between energy diffusion and the butterfly velocity [16,17,[34][35][36][37][38]. Furthermore, the dual role of σ as the mode for "propagating chaos" and energy conservation leads to a new phenomenon called "pole-skipping" [21,22] in the energy-density two-point function, which has been confirmed in a variety of maximally chaotic systems including general holographic systems dual to Einstein gravity and its higher derivative extensions [19,20,23,24].
Let us now elaborate a bit more on (2.6)-(2.9), and how to use the above elements to compute four-point functions of generic few-body operators. We can expand L (1) as where c mn are constants. Then equation (2.7) can be written more explicitly as (2.14) Four-point functions of generic few-body operators can be computed at leading order in and the second is an OTOC 6 Throughout this paper we use the notation O = Tr(Oρ 0 ) with ρ 0 = e −β0H /Z the thermal density matrix and take V, W to be Hermitian.
with t 3,4 t 1,2 . Using (2.6) we find (2.15) can be written as 7 where g V , g W are two point functions of the bare operatorsV ,Ŵ , t 12 = t 1 − t 2 , and in the above equation it should be understood that L (1) 's also act on g V or g W after them. G ± are Wightman functions 8 of (t), with c ± some constants proportional to 1/N , and c a real constant. Note that in the above expressions we have only made manifest the exponential terms in t. A similar expression can be obtained for H 4 . Note that One can check that due to (2.7)-(2.9), the exponentially growing terms are canceled in TOC (2.17). Then due to (2.20) and (2.19) one finds that OTOC H 4 must contain an exponentially growing piece.

III. CONSISTENCY OF EFT REQUIRES MAXIMAL CHAOS
In this section we present a simple argument to show that the consistency of the shift symmetry of the effective vertex and fluctuation-dissipation relations of general operators require that the Lyapunov exponent be maximal.
In particular the two-point function g V of V should satisfy the fluctuation-dissipation theorem (FDT), which can be used to constrain λ through (2.11). Applying FDT to g V in 7 Note that our definitions of G 4 , H 4 differ slightly from those in [21] as we are not normalising them by the bare correlation function. 8 For λ = λ max there are also additional exponential terms in G ± of the form te λt .
Assuming that equation (2.11) can be analytically continued to the range Im t ∈ [0, β 0 ), we We therefore conclude that these conditions are only consistent for a maximal Lyapunov exponent, and that the hydrodynamic theory proposed in [21] is a theory of maximally chaotic systems.
A further connection to recent discussions of maximal chaos can be seen by noting that one can use the shift symmetry (2.11) and (2.14) to write the exponentially growing behaviour in H 4 in a particularly simple form WhereF even (λ, t) is analogous to F even (λ, t) defined above, but is defined using g W (t). The expression (3.3) matches an ansatz for the OTOC of 0+1 dimensional systems proposed in [14,31] in the case λ = λ max .

SYSTEMS
In this section we first present a general argument showing that in the chaos EFT, the commutator square (1.1) always vanishes at leading order. We then support the conclusion with another argument, and illustrate it using the examples of SYK model and holographic systems. In particular we note that in holographic systems the commutator square is always determined by stringy corrections.

A. EFT derivations
Now that we have shown that the shift symmetry requires maximal chaos, here we present a new prediction of this theory for the commutator square We are interested in the regime t 3 , t 4 t 1 , t 2 . The double commutator consists of two time ordered and two out-of-time ordered correlation functions, and H 4 ,H 4 out-of-time ordered .
Likewise it is straightforward to show that The vanishing of (4.1) at leading order can be intuitively understood from Fig. 3: fourpoint functions all reduce to two-point functions of σ, while for (4.1) to be non-vanishing one needs four-point function of σ, which are higher orders in 1/N .
FIG. 3. At leading order in large N OTO four-point functions defined on a four-contour reduce to a sum of two-point functions of σ(t). They can therefore be calculated from the effective action of the hydrodynamic field σ on a CTP contour.
We can also consider commutators which are separated in imaginary time, such as where in order for the double commutator to be defined it is necessary to use a configuration where imaginary parts of t 2 , t 4 match and likewise for t 1 , t 3 . The same arguments as above show that C θ 4 (t 1 , t 2 , t 3 , t 4 ) = 0 at leading order in 1/N . While we have focused on (0 + 1)-dimensional systems for simplicity, we expect that in higher dimensional systems, the argument for the exact cancellation of the chaotic mode contribution to the commutator square [W (t 4 , x 4 ), V (t 2 , x 2 )][W (t 3 , x 3 ), V (t 1 , x 1 )] also applies. Explicit examples of higher-dimensional chaos EFT include those for SYK chains [16] and CFT constructions [19,20].
Known examples of maximal chaos all happen in some limit, in addition to N → ∞. For the SYK model this corresponds to the low temperature limit, for holographic systems it is the classical gravity limit (or infinite coupling), and for CFTs in a Rindler wedge it is the light-cone limit. In such a limit, the stress tensor exchange gives the dominant contribution to OTOCs, and is responsible for the appearance of maximal chaos. For convenience of discussion we will denote the maximal chaos limit in these theories collectively as the g → ∞ limit, with g standing for the corresponding parameter in each system. Given that the contribution of the stress tensor exchange to the commutator square (4.1) vanishes, the leading order result is then given by contributions from other modes. We will discuss in Sec. IV C the examples of SYK and holographic systems in detail. In the conclusion Sec. VI we also briefly speculate on the physical interpretation of the vanishing of the commutator square at leading order. Here we will briefly comment on the general features.
Including the contributions from an infinite number of other modes leads to two main corrections to the behavior of the OTOC. Firstly, the Lyapunov exponent is shifted where c 1 > 0 and α 1 > 0 are some constants which depend on specific systems. Secondly, the prefactor to e λt in the OTOC also receives high order corrections in 1/g. where C(g) is real. Plugging in the above expression into the commutator square 9 we then find that to leading order in the large g limit, (4.11) The leading order behavior of the prefactor in the above equation depends on how C(g) behaves as g → ∞. For example, if C is finite as g → ∞ then the prefactor vanishes as g −α 1 in the large g limit.

B. Other arguments
Before discussing explicit examples, for completeness we mention some other arguments which suggest that the commutator square vanishes for maximal chaos at leading order.
More generally, in [14,31] an ansatz was proposed for the leading order in 1/N exponential growth of the OTOC correlation function where the overall prefactor C ∼ 1/N is real, and Υ R WW (t 34 ), Υ A VV (t 12 ) are advanced and retarded vertex functions that satisfy fluctuation-dissipation relations. The phase factor e iλβ 0 /4 is consistent with (4.13) being real. For λ = 2π/β 0 , equation (4.15) is also consistent with the form (3.3) from the effective theory. It then follows from (4.15) that  Again for maximal chaos the cos(λβ 0 /4) factor suppresses the leading contribution to C θ 4 .

C. Examples
We now use the examples of SYK model and holographic systems to illustrate the somewhat abstract discussion above. A suppression of exponential growth in the commutator square of various maximally chaotic systems has previously been observed in [4,14,29,31].

SYK model
The ansatz (4.15) for a general system was in fact motivated in large part by the form of the OTOC in soluble (i.e. strong coupling or large-q) limits of the SYK model. For the 10 Note F (t) is real.
Note that the leading order result for H 4 in the large β 0 J limit can also be obtained from the Schwarzian theory, which provides a specific example of the hydrodynamic effective theory discussed in Section II. From the argument in (4.7) we expect that the non-exponential parts of the commutator square should also cancel identically in the Schwarzian theory. This can indeed be confirmed explicitly from analytically continuing the results in [10].

Holographic systems
As another example, let us consider the behavior of the commutator square in higher dimensional holographic theories, both in classical gravity and in string theory. Earlier discussion includes [4,30]. For concreteness we consider the out-of-time-ordered correlator where β 0 , and the corresponding commutator square At the level of the gravity approximation, which corresponds to N → ∞ and λ h → ∞ in the boundary theory 11 , the exponential growth of H 4 has the form [2,4] where c ∼ O(1) is real. The expression (4.24) comes from graviton exchanges which translates into the boundary theory from exchanges of the stress tensor. Such exchanges are fully captured by our EFT and we thus expect that in the classical gravity including the non-exponential parts.
In Appendix A we show that within the eikonal approximation the sum of H 4 andH 4 vanishes identically in any theory dual to classical gravity. As such, there is no exponential growth in C 4 to leading order in 1/N in classical gravity. In particular, we find that this cancellation is a direct consequence of the fact the phase-shift for the bulk gravitational scattering process is real. The suppression of the commutator OTOC in gravitational systems can therefore be seen to be a consequence of bulk scattering being elastic.
The above considerations hold for any system dual to a classical gravity theory, even if the bulk theory contains higher derivative terms reflecting certain finite coupling corrections. 11 For example, when the boundary theory is the N = 4 Super-Yang-Mills theory, λ h is the 't Hooft coupling.
In order to find a non-vanishing expression for C 4 at leading order in 1/N , it is necessary to include not just the effects of gravitational scattering but also the exchange of stringy modes at finite string coupling α . This translates into the boundary theory as exchanges of an infinite number of other intermediate operators (including in particular high-spin operators) in addition to the stress tensor.
The effect of such stringy corrections on the OTOC has been discussed in detail in [4]. The discussion is a bit technical, however the qualitative features of these results are illuminating.
The important result is that H 4 is controlled by the phase shift of a bulk scattering process.
In particular, the functional form of the OTOC shows a sharp crossover between a stringy regime for which | x|/t < v * and a graviton dominated regime for | x|/t > v * where for the black hole background studied in [4,30] we have with R the AdS curvature radius and d the number of spacetime dimensions in the boundary theory.
To elaborate, for short distances | x| < t/v * , where v * is some specific velocity, H 4 can be computed from an integral representation by a saddle point approximation and stringy corrections are important. In this regime it takes the form where f 1 (| x|/t) is complex, the string corrected Lyapunov exponent λ is given by (4. 28) and the constant D is given by In the regime | x| > t/v * even at finite α , the integral for H 4 is dominated by a graviton pole. As a result the contribution from graviton exchange dominates, and H 4 is given simply by the classical gravity result (4.24).
Here we point out that, for C 4 (4.23), such a crossover does not exist, and its behavior is controlled by stringy exchanges for all t and x. This can be intuitively expected from that the gravitational contribution to C 4 vanishes identically to leading order in 1/N . We give technical details in Appendix A. There we show that the form of C 4 at large N is controlled by the imaginary part of the phase shift for a bulk scattering, which is zero for graviton exchange, but non-zero due to the effects of stringy modes. As a result, there is no graviton pole in the integral expression for C 4 and there is no cross-over in its behavior.
More explicitly, as shown in Appendix A, for all | x|/t, C 4 has the form For | x|/t < v * , there is a suppression in the prefactor compared with (4.27) which can be through of as a higher dimensional counterpart of the cos(λβ 0 /4) factor seen 0 + 1 dimensional systems 12 .

V. A SIMPLE ARGUMENT FOR THE CROSSOVER VELOCITY
As we have just discussed, in addition to the above example of holographic systems at finite α , it has been observed in various other systems that there can exist a crossover velocity v * beyond which maximally chaotic behavior is found even if theory is non-maximally chaotic for small v [16,29]. A more elaborate discussion of this crossover, and further examples of such systems, can be found in [30]. In this section we consider a simplest scenario for the existence of such a crossover velocity 13 .
Consider higher spin exchange contribution to H 4 defined in (4.22). At distances | x| β 0 we expect the OTOC should decay exponentially with | x|. In other words, the time ordering should not affect the validity of cluster decomposition principle at large distances. Denoting 12 Note cos(λβ 0 /4) = sin(πd(d − 1)α /8R 2 ) and hence (4.31) reduces to 2 cos(λβ 0 /4) fior v = 0. 13 We thank discussion with Gabor Sarosi and Mark Mezei for the content of this section.
the contribution of a spin-s operator to H 4 as A s , we then expect where M (s, β 0 ) is a function of spin s and the inverse temperature β 0 . M (s, β 0 ) will also depend on any coupling constants in theory (e.g. α in string theory) and may also depend on other quantum numbers of the exchange operator. For example, for a two-dimensional CFT in the large central charge limit, one finds that [3] M (s, where ∆ is the conformal dimension of the operator.
Recall that the contribution of the stress tensor can be written as whereṽ B is not necessarily the butterfly velocity of the full system. Comparing (5.1) with (5.3) we see that A s dominates over that from the stress tensor for a given | x| = vt if If (5.4) is satisfied for one s > 2 it must be satisfied for an infinite number of them, otherwise the chaos bound will be violated. Notice that the above equation is always satisfied for all s > 2 for v = 0, which is consistent with the fact that upon including finite-coupling corrections SYK models and holographic systems will be non-maximally chaotic at v = 0 [4,16].
A simplest scenario for the existence a crossover velocity v * is the existence of a critical velocity v c such that for v > v c , equation ( can be considered as defining a finite temperature version of the "leading Regge trajectory." Then existence of such a v c then implies that (5.5) In particular, the left hand side must have a finite limit as s → ∞ v ∞ can of course be larger or smaller than v * . The above equation implies that M 0 (s, β 0 ) should increase with s at least as fast as linear dependence. In the case that v ∞ is nonzero, M 0 must also be linear in s .
We stress that the above discussion is only a simplest scenario. It can happen that v * exists without existence of v c or v ∞ .

VI. DISCUSSION
In this paper we have discussed various features of a maximally chaotic system, motivated by the chaos effective field theory introduced in [21]. It is also of interest to better understand the physical interpretation of the vanishing of the leading order contribution to the commutator square for maximally chaotic systems.
It seems likely this vanishing indicates a sharp distinction between the nature of operator scrambling in maximally and non-maximally chaotic systems. One natural interpretation is that this vanishing suggests an extra "unitarity" constraint on the scrambling processes of maximally chaotic systems. Heuristically, if we view an OTOC as a scattering amplitude, the vanishing of the commutator square can be interpreted as the absence of "particle" production for the scattering process, i.e. the scattering is elastic. This can be made precise in various contexts. In the gravity description given in Appendix A, the phase shift for the corresponding string scattering is real in the maximally chaotic regime. Similarly, for Rindler OTOC, the vanishing of the commutator square means the vanishing of the imaginary part of the CFT scattering amplitude corresponding to the correlator [40]. These precise statements resonate with the discussion of scrambling in generic 0+1 dimensional systems in [14,31], which used a restricted Hilbert space to define a scattering matrix that becomes unitary in the case of maximal chaos 14 . Further, the existence of an enhanced "unitarity" for maximal chaos is related to the discussion of [29], where the coefficient of exponential growth in commutator OTOCs is inversely proportional to the branching time. As such this suggests that branching processes are absent in systems for which commutator squares vanish.
slightly more general set of correlation functions than discussed in Section IV C 2, Recall that H 4 defined above can be expressed as [4] where the integral runs over x, x , p u 1 , p v 2 and the parameter a 0 is simply a constant. Further, δ(s, b) is the phase shift of a bulk two-to-two scattering process and is expressed as a function of the centre of mass energy s = a 0 p u 1 p v 2 and transverse separation b = | x− x | of the scatterred quanta. The various ψ i in (A1) are Fourier transforms of bulk-to-boundary propagators 15 with φ V , φ W the dual bulk fields dual to Hermitian boundary operators V, W . Using the identitỹ to computeH 4 we find that the commutator C θ 4 = H 4 +H 4 + . . . can be written as 16 15 Note that the labelling of fields and operators in (A1) is slightly different to the corresponding formula in [4] as we are considering a slightly different arrangement of operators. 16 Here we are only interested in the exponential growth of C θ 4 , so are ignoring the time-ordered pieces with the bulk to boundary propagators again given by the formulae in (A2). In both string theory and classical gravity δ ∼ G N ∼ 1/N , and hence an expansion in 1/N amounts to expanding the exponentials in (A1) and (A3). To leading order 1/N we therefore find For theories dual to classical gravity δ(s, b) is always purely real and can be computed by evaluating the on-shell gravitational action of a pair of gravitational shock waves [2,4]. It is immediately clear from (A4) that all C θ vanish identically to leading order in 1/N in such theories within the eikonal approximation.
To see the behavior of (A6) let us first recall the evaluation of (A5) [4]. With y ≡ vt and the explicit form of s plugged in, equation (A5) can be written as where v = | v|. At large t the integral in (A7) has a saddle-point at k * where with v B the butterfly velocity. For v < v * the integral can be computed using the saddle-point approximation. The resulting expression for the phase-shift δ(t, v) evaluates to where λ(v) is the velocity dependent Lyapunov exponent [29,39] where in obtaining these expressions we have eliminated µ and r 0 using that for the black hole background studied in [4] we have v B = d/2(d − 1)) and r 0 = 4πRβ −1 0 d −1 with R the AdS radius. Recalling that the bulk to boundary wavefunctions in (A1) are spatially peaked around the positions of the external operators we have from (A4) that the functional dependence of (4.22) is given at leading order in 1/N by iδ(t, v) with v = | x|/t. This gives rise to the functional form of (4.27) discussed in the main text.
Note that the expression (A10) diverges at v = v * , which can be attributed to the existence of a graviton pole at k 2 + µ 2 = 0 in the integrand of (A7). For v > v * the contribution from the graviton pole dominates and leads to the behavior of maximal chaos.
In contrast, there is no graviton pole in (A6) due to the factor of sin(πα ( k 2 + µ 2 )/4r 2 0 ) in the numerator. So in this case, the saddle-point approximation is valid for all x and t.