MMIT-CTP/5283
February 24, 2021
On systems of maximal quantum chaos
Mike Blake and Hong Liu School of Mathematics, University of Bristol, Fry Building,Woodland Road, Bristol BS8 1UG, UK Center for Theoretical Physics, Massachusetts Institute of Technology,Cambridge, MA 02139
Abstract
A remarkable feature of chaos in many-body quantum systems is the existence of a bound onthe quantum Lyapunov exponent. An important question is to understand what is special aboutmaximally 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 sys-tems. We first provide evidence that a hydrodynamic effective field theory of chaos we previouslyproposed should be understood as a theory of maximally chaotic systems. We then emphasize andmake explicit a signature of maximal chaos which was only implicit in prior literature, namely thesuppression of exponential growth in commutator squares of generic few-body operators. We pro-vide a general argument for this suppression within our chaos effective field theory, and illustrate itusing SYK models and holographic systems. We speculate that this suppression indicates that thenature of operator scrambling in maximally chaotic systems is fundamentally different to scram-bling in non-maximally chaotic systems. We also discuss a simplest scenario for the existence of amaximally chaotic regime at sufficiently large distances even for non-maximally chaotic systems. a r X i v : . [ h e p - t h ] F e b ONTENTS
I. Introduction 2II. A quick review of the EFT for maximal chaos 5III. Consistency of EFT requires maximal chaos 9IV. Suppression of commutator square for maximally chaotic systems 10A. EFT derivations 11B. Other arguments 13C. Examples 141. SYK model 142. Holographic systems 16V. A simple argument for the crossover velocity 18VI. Discussion 20Acknowledgements 21A. Formula for commutator square in holographic systems 21References 25
I. INTRODUCTION
One of the key dynamical concepts characterizing a quantum many-body system is scram-bling of quantum information. Suppose we slightly disturb a system by inserting a local few-body 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 infor-mation of the original disturbance is scrambled among more and more degrees of freedom.2hile for a general many-body system such evolutions are extremely complicated, it hasbeen increasingly recognized during the last decade that there are remarkable universalitiesin this process [1–15]. For example, out-of-time-ordered correlators (OTOCs) have emergedas important probes of scrambling in chaotic systems. In particular, in chaotic systems witha large number of single-site degrees of freedom, the commutator squares of generic few-bodyoperators grow exponentially with time at early times (below N characterizes the numberof single-site degrees of freedom) (cid:10) [ W ( t ) , V (0)] (cid:11) ∝ N e λt (1.1)with a quantum Lyapunov exponent λ bounded by [5] λ ≤ λ max = 2 π (cid:126) β . (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 holographicsystems, SYK and its variations, two-dimensional CFTs with a large central charge (and agap), 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 hasemerged in such systems is that finite temperature dynamics of stress tensor appears tounderlie the observed maximally chaotic behavior. In other words, the chaotic behaviorappears to be “hydrodynamic” in nature.It is tempting to use the hydrodynamic nature of chaos as a defining feature of maximallychaotic systems. In this paper we would like to present further support for this idea, and todiscuss 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 For notational consistency with [21], we denote the inverse temperature by β . Henceforth we set (cid:126) = 1. . In particular we give a general argument showing that in holographic theories thecommutator square is determined at leading order in 1 / N by stringy corrections .It has been observed in various examples that even non-maximally chaotic systems couldexhibit maximally chaotic behavior at sufficiently large distances [4, 16, 29, 30]. For suchsystems, the chaos EFT can apply at large distances. We discuss a simplest scenario forsuch a phenomenon.The plan of the paper is as follows. In Sec. II we give a quick review of key features ofthe 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 discussa simplest scenario for the existence of a maximally chaotic regime at sufficiently largedistances for non-maximally chaotic systems. Sec. VI concludes with a brief discussion offuture questions. Appendix A contains some technical details for Sec. IV. The suppression of exponential growth in commutator squares in these models is implicit in the resultspresented in [4, 14, 29, 31]. The importance of stringy corrections in determining the form of the commutator square in SYK modelsis discussed in [29]. I. A QUICK REVIEW OF THE EFT FOR MAXIMAL CHAOS
In this section we present a quick review of the chaos EFT introduced in [21], whichis based on the following elements (to avoid cluttering notations below we first use 0 + 1dimensional systems as an illustration):1. One imagines the scrambling of a generic few-body operator V ( t ) at a finite temper-ature allows a coarse-grained description in terms of building up a “hydrodynamiccloud,” i.e. we can write V ( t ) = V [ ˆ V , σ ( t )] . (2.1)Here ˆ V ( 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 de-grees of freedom. Bare operators which involve different degrees of freedom have nocorrelations, i.e. (cid:68) ˆ V ˆ W (cid:69) = 0 for generic V (cid:54) = W .2. One identifies the chaos mode σ with the collective degrees of freedom associated withenergy conservation, and thus its dynamics is governed by hydrodynamics. However,in order to capture the growth of V ( t ), one needs a “quantum hydrodynamics” whichis valid in the regime ∆ t ∼ β , with ∆ t typical time scales of interests. This quantumhydrodynamic theory can be considered as a generalization of the conventional hydro-dynamics without doing a derivative expansion , and is non-local. Its action, whichalso incorporates dissipative effects, can be written using the techniques developedin [32, 33]. The explicit form of the action S EFT [ σ ] is not important for our discus-sion below (we refer interested readers to [21] for details) except that the Lagrangiandepends on σ only through derivatives and the equilibrium solution is given by σ eq ( t ) = t . (2.2)3. To describe a chaotic system, we still need to impose an additional shift symmetry e − λσ → e − λσ + c (2.3) Recall that conventionally hydrodynamics is formulated as a derivative expansion and is valid for ∆ t (cid:29) β . V WW FIG. 1. At leading order in large N correlation functions are controlled by exchange of hydrody-namic fields σ ( t ). on the action S EFT [ σ ] and on the coupling (2.1) of σ to general few-body operators.Here c is an arbitrary positive constant and λ is the quantum Lyapunov exponent. Nearequilibrium we introduce the deviation of σ ( t ) from the equilibrium solution (2.2) (cid:15) ( t ) = σ ( t ) − t (2.4)For infinitessimal (cid:15) ( t ) the symmetry (2.3) implies a symmetry under (cid:15) → (cid:15) − ce λt . (2.5)This shift symmetry results in an exponential growing piece in two-point functions of σ which is the origin of chaotic behaviour within the EFT. Nevertheless the EFT descrip-tion of σ appears to be self-consistent for any value of λ . In particular, stress-tensorcorrelation functions do not contain an exponentially growing piece, and fluctuation-dissipation 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),fluctuation-dissipation relations of the operator require λ to take the maximal value (1.2).4. Using S EF T [ σ ] and the coupling (2.1) one can calculate correlation functions of genericoperators perturbatively in 1 / N . To leading order in 1 / N it is sufficient to considertree-level exchange of σ as in Fig. 1. We require the coupling (2.1) between σ and ageneral operator be also invariant under the shift symmetry (2.3). Near equilibrium,6quation (2.1) can be written using (2.4) in a form V ( t ) = ˆ V ( t ) + L (1) [ ˆ V (cid:15) ]( t ) + O ( (cid:15) ) (2.6)where L (1) is some differential operator acting on both ˆ V and (cid:15) . Invariance under theshift symmetry implies the condition L (1) t [ g V ( t ) e λt ] + L (1) t [ g V ( t ) e λt ] = 0 (2.7)where subscript t in L (1) t refers to the variable of the derivatives and g V ( t ) is thetwo-point function of ˆ V g V ( t ) = (cid:104) ˆ V (0) ˆ V ( t ) (cid:105) = G ˆ V − ( t ) (2.8)It was also found in [21] that in order for the exponentially growing behavior to cancelfor all possible configurations of TOC it is also necessary to impose the version of (2.7)with λ → − λ , i.e. L (1) t [ g V ( t ) e − λt ] + L (1) t [ g V ( t ) e − λt ] = 0 . (2.9)The shift symmetry may be considered as an emergent “macroscopic” symmetry. It hasmany implications: (i) constrains the structure of correlation functions to all derivative or-ders; (ii) implies existence of an exponentially growing mode the exchange of which leads toexponential growth of OTOCs; (iii) leads to a connection between energy diffusion and thebutterfly velocity [16, 17, 34–38]. Furthermore, the dual role of σ as the mode for “propagat-ing 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 maximallychaotic systems including general holographic systems dual to Einstein gravity and its higherderivative extensions [19, 20, 23, 24].Let us now elaborate a bit more on (2.6)–(2.9), and how to use the above elements tocompute four-point functions of generic few-body operators. We can expand L (1) as L (1) [ ˆ V (cid:15) ] = ∞ (cid:88) n,m =0 c nm ∂ nt ˆ V ∂ mt (cid:15) + O ( (cid:15) ) (2.10) Here by “macroscopic” we refer to scales of order β . V ( t ) V ( t ) W ( t ) W ( t ) ⇢ V ( t ) V ( t ) W ( t ) W ( t ) FIG. 2. Left: operator insertions for (2.15). Right: operator inserations for (2.16). where c mn are constants. Then equation (2.7) can be written more explicitly as F even ( λ, t ) F odd ( λ, t ) = − tanh λt . (2.11)where we have defined F even ( λ, t ) = (cid:88) n even f n ( λ ) ∂ nt g V ( t ) , F odd ( λ, t ) = (cid:88) n odd f n ( λ ) ∂ nt g V ( t ) . (2.12)with f n = (cid:88) m c nm λ m . (2.13)Similarly (2.9) implies F even ( − λ, t ) F odd ( − λ, t ) = tanh λt . (2.14)Four-point functions of generic few-body operators can be computed at leading order in1 / N expansion by tree-level exchange of σ as indicated in Fig. 1. For illustration, considertwo explicit examples indicated in Fig. 2. The first is a TOC G ( t , t , t , t ) = (cid:104) V ( t ) W ( t ) W ( t ) V ( t ) (cid:105) (2.15)and the second is an OTOC H ( t , t , t , t ) = (cid:104) W ( t ) V ( t ) W ( t ) V ( t ) (cid:105) (2.16) Throughout this paper we use the notation (cid:104) O (cid:105) = Tr(O ρ ) with ρ = e − β H /Z the thermal density matrixand take V, W to be Hermitian. t , (cid:29) t , . Using (2.6) we find (2.15) can be written as G − g V ( t ) g W ( t ) = (cid:16) L (1) t L (1) t G + ( t ) + L (1) t L (1) t G + ( t )+ L (1) t L (1) t G − ( t ) + L (1) t L (1) t G − ( t ) (cid:17) g V ( t ) g W ( t ) (2.17)where g V , g W are two point functions of the bare operators ˆ V , ˆ W , t = t − t , and in theabove equation it should be understood that L (1) ’s also act on g V or g W after them. G ± areWightman functions of (cid:15) ( t ), G + ( t ) = (cid:104) (cid:15) ( t ) (cid:15) (0) (cid:105) = c + e λt + · · · G − ( t ) = (cid:104) (cid:15) (0) (cid:15) ( t ) (cid:105) = c − e λt + · · · , (2.18)∆( t ) = (cid:104) [ (cid:15) ( t ) , (cid:15) (0)] (cid:105) = − ice λt + · · · , c = i ( c + − c − ) (2.19)with c ± some constants proportional to 1 / N , and c a real constant. Note that in the aboveexpressions we have only made manifest the exponential terms in t . A similar expressioncan be obtained for H . Note that H − G = (cid:104) [ W ( t ) , V ( t )] W ( t ) V ( t ) (cid:105) = L (1) t L (1) t [ g V ( t ) g W ( t )∆( t )] (2.20)One can check that due to (2.7)–(2.9), the exponentially growing terms are canceled inTOC (2.17). Then due to (2.20) and (2.19) one finds that OTOC H must contain anexponentially growing piece. III. CONSISTENCY OF EFT REQUIRES MAXIMAL CHAOS
In this section we present a simple argument to show that the consistency of the shiftsymmetry of the effective vertex and fluctuation-dissipation relations of general operatorsrequire that the Lyapunov exponent be maximal.In particular the two-point function g V of V should satisfy the fluctuation-dissipationtheorem (FDT), which can be used to constrain λ through (2.11). Applying FDT to g V in Note that our definitions of G , H differ slightly from those in [21] as we are not normalising them by thebare correlation function. For λ = λ max there are also additional exponential terms in G ± of the form te λt . g V ( t ) = g V ( iβ − t ) (for Im t ∈ [0 , β )) which in turn leads to F even ( λ, t ) = F even ( λ, iβ − t ) , F odd ( λ, t ) = − F odd ( λ, iβ − t ) . (3.1)Assuming that equation (2.11) can be analytically continued to the range Im t ∈ [0 , β ), wefind that it is compatible with (3.1) only iftanh λt λ ( t − iβ )2 = ⇒ λ = 2 πβ (3.2)We therefore conclude that these conditions are only consistent for a maximal Lyapunovexponent, and that the hydrodynamic theory proposed in [21] is a theory of maximallychaotic systems.A further connection to recent discussions of maximal chaos can be seen by noting that onecan use the shift symmetry (2.11) and (2.14) to write the exponentially growing behaviourin H in a particularly simple form H − g V ( t ) g W ( t ) = ic F even ( − λ, t )sinh λt ˜ F even ( λ, t )sinh λt e λ ( t + t − t − t ) / + · · · . (3.3)Where ˜ F even ( λ, t ) is analogous to F even ( λ, t ) defined above, but is defined using g W ( t ). Theexpression (3.3) matches an ansatz for the OTOC of 0+1 dimensional systems proposed in[14, 31] in the case λ = λ max . IV. SUPPRESSION OF COMMUTATOR SQUARE FOR MAXIMALLY CHAOTICSYSTEMS
In this section we first present a general argument showing that in the chaos EFT, thecommutator square (1.1) always vanishes at leading order. We then support the conclusionwith another argument, and illustrate it using the examples of SYK model and holographicsystems. In particular we note that in holographic systems the commutator square is alwaysdetermined by stringy corrections. 10 . EFT derivations
Now that we have shown that the shift symmetry requires maximal chaos, here we presenta new prediction of this theory for the commutator square C ( t , t , t , t ) = (cid:104) [ W ( t ) , V ( t )][ W ( t ) , V ( t )] (cid:105) . (4.1)We are interested in the regime t , t (cid:29) t , t . The double commutator consists of two timeordered and two out-of-time ordered correlation functions, C ( t , t , t , t ) = H + ˜ H − G − ˜ G , (4.2)with G , ˜ G time ordered G = (cid:104) V ( t ) W ( t ) W ( t ) V ( t ) (cid:105) ˜ G = (cid:104) W ( t ) V ( t ) V ( t ) W ( t ) (cid:105) , (4.3)and H , ˜ H out-of-time ordered H = (cid:104) W ( t ) V ( t ) W ( t ) V ( t ) (cid:105) ˜ H = (cid:104) V ( t ) W ( t ) V ( t ) W ( t ) (cid:105) . (4.4)Using equations (3.3), (3.1), ˜ H ( t , t , t , t ) = H ( t + iβ , t , t , t ) and that λ = λ max onefinds that the exponential pieces in H and ˜ H precisely cancel, i.e. C ( t , t , t , t ) does nothave any exponentially growing pieces. In fact, the contribution from tree-level exchange of σ to the double commutator in (4.1) is identically zero. Recall from (2.20) H − G = L (1) t L (1) t [ g V ( t ) g W ( t )∆( t )] , ∆( t ) = (cid:104) [ (cid:15) ( t ) , (cid:15) ( t )] (cid:105) . (4.5)Likewise it is straightforward to show that˜ H − ˜ G = − L (1) t L (1) t [ g V ( t ) g W ( t )∆( t )] = − ( H − G ) , (4.6)= ⇒ C ( t , t , t , t ) = 0 , at leading order in 1 N . (4.7)The vanishing of (4.1) at leading order can be intuitively understood from Fig. 3: four-point functions all reduce to two-point functions of σ , while for (4.1) to be non-vanishingone needs four-point function of σ , which are higher orders in 1 / N .11 ( t i ) ( t j ) ⇢ = X i,j ⇢ V ( t ) V ( t ) W ( t ) W ( t ) FIG. 3. At leading order in large N OTO four-point functions defined on a four-contour reduce toa sum of two-point functions of σ ( t ). They can therefore be calculated from the effective action ofthe hydrodynamic field σ on a CTP contour. We can also consider commutators which are separated in imaginary time, such as C θ ( t , t , t , t ) = (cid:104) [ W ( t + iθ ) , V ( t + iθ )][ W ( t ) , V ( t )] (cid:105) (4.8)where in order for the double commutator to be defined it is necessary to use a configurationwhere imaginary parts of t , t match and likewise for t , t . The same arguments as aboveshow that C θ ( t , t , t , t ) = 0 at leading order in 1 / N .While we have focused on (0 + 1)-dimensional systems for simplicity, we expect that inhigher dimensional systems, the argument for the exact cancellation of the chaotic modecontribution to the commutator square (cid:104) [ W ( t , (cid:126)x ) , V ( t , (cid:126)x )][ W ( t , (cid:126)x ) , V ( t , (cid:126)x )] (cid:105) also ap-plies. 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 → ∞ . Forthe SYK model this corresponds to the low temperature limit, for holographic systems it isthe classical gravity limit (or infinite coupling), and for CFTs in a Rindler wedge it is thelight-cone limit. In such a limit, the stress tensor exchange gives the dominant contributionto OTOCs, and is responsible for the appearance of maximal chaos. For convenience ofdiscussion 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 thecontribution of the stress tensor exchange to the commutator square (4.1) vanishes, theleading order result is then given by contributions from other modes. We will discuss in12ec. IV C the examples of SYK and holographic systems in detail. In the conclusion Sec. VIwe also briefly speculate on the physical interpretation of the vanishing of the commutatorsquare 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 maincorrections to the behavior of the OTOC. Firstly, the Lyapunov exponent is shifted λ = 2 πβ − c g α , g → ∞ (4.9)where c > α > e λt in the OTOC also receives high order corrections in 1 /g . For concretenessconsider the configuration t = 0, t = i(cid:15) , t = t , t = t + i(cid:15) with (cid:15) (cid:28) β . Then in theexamples of SYK and holographic systems [4, 14, 29, 31], the OTOC has the form H ( t ) = iC ( g ) e − ic β gα e λt = C ( g ) e iλβ / e λt (4.10)where C ( g ) is real. Plugging in the above expression into the commutator square we thenfind that to leading order in the large g limit, C (cid:15) ( t ) = Cc β g α e πβ t , g → ∞ . (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 in the large g limit. B. Other arguments
Before discussing explicit examples, for completeness we mention some other argumentswhich suggest that the commutator square vanishes for maximal chaos at leading order.Consider first the commutator square C − β / ( t ) corresponding to (4.8) with t = t = 0 and t = t = t and θ = − β /
2. This can be written as [5] C − β / ( t ) = F ( t + iβ /
4) + F ( t − iβ /
4) + · · · (4.12) Note for this configuration C (cid:15) ( t ) = 2 Re( H ( t )) + . . . with the dots indicating non-exponential pieces. F ( t ) the thermally regulated OTOC where operators are symmetrically placed aroundthe unit circle F ( t ) = Tr (cid:18) W ( t ) ρ / V (0) ρ / W ( t ) ρ / V (0) ρ / (cid:19) . (4.13)Assuming F ( t ) ∼ Ce λt with real C ∼ / N we then have C − β / ( t ) = 2 C cos( λβ / e λt + · · · . (4.14)For maximal chaos we find a suppression as the prefactor factor cos( λβ /
4) vanishes.More generally, in [14, 31] an ansatz was proposed for the leading order in 1 / N exponentialgrowth of the OTOC correlation function H ( t , t , t , t ) = Ce iλβ / e λ ( t + t − t − t ) / Υ R WW ( t )Υ A VV ( t ) (4.15)where the overall prefactor C ∼ / N is real, and Υ R WW ( t ), Υ A VV ( t ) are advanced andretarded vertex functions that satisfy fluctuation-dissipation relations. The phase factor e iλβ / is consistent with (4.13) being real. For λ = 2 π/β , equation (4.15) is also consistentwith the form (3.3) from the effective theory. It then follows from (4.15) that C θ ( t , t , t , t ) = 2 C cos( λβ / e λ ( t + t − t − t ) / Υ A ( t − iθ )Υ R ( t − iθ ) . (4.16)Again for maximal chaos the cos( λβ /
4) factor suppresses the leading contribution to C θ . C. Examples
We now use the examples of SYK model and holographic systems to illustrate the some-what abstract discussion above. A suppression of exponential growth in the commutatorsquare of various maximally chaotic systems has previously been observed in [4, 14, 29, 31].
1. SYK model
The ansatz (4.15) for a general system was in fact motivated in large part by the formof the OTOC in soluble (i.e. strong coupling or large- q ) limits of the SYK model. For the Note F ( t ) is real. g in (1.1) can be identified as g = β J and g → ∞ correspondsto the low temperature limit. In this case [8, 29] λ = 2 πβ (cid:18) − cβ J (cid:19) (4.17)i.e. α = 1 in (4.9) and C ∼ β J N . We thus find C cos( λβ / ∼ O (( β J ) ) 1 N (4.18)which is suppressed by a factor 1 / ( β J ) compared with OTOC. These features can be illus-trated explicitly in the large- q limit, for which a closed form expression of H for all β J wasobtained in [29]. The exponential growing part of H can be written as H ≈ N cos( νπ/ e πνβ ( t + t − t − t + iβ / (2cosh( πνβ ( t − iβ ))(2cosh( πνβ ( t − iβ ))) (4.19)where v ∈ (0 ,
1) parametrises the strength of the coupling β J through πν cos( πν/
2) = β J . (4.20)In the large β J limit, ν ≈ H is proportional to β J N . In contrast, thecommutator OTOC is given by C ≈ N e πνβ ( t + t − t − t ) (2cosh( πνβ ( t − iβ ))(2cosh( πνβ ( t − iβ ))) (4.21)As indicated above the prefactor of the exponential growth is suppressed relative to H bya factor 2 cos( πν/
2) = 2 cos( λβ / β J → ∞ .Note that the leading order result for H in the large β J limit can also be obtainedfrom the Schwarzian theory, which provides a specific example of the hydrodynamic effectivetheory discussed in Section II. From the argument in (4.7) we expect that the non-exponentialparts of the commutator square should also cancel identically in the Schwarzian theory. Thiscan indeed be confirmed explicitly from analytically continuing the results in [10].15 . Holographic systems As another example, let us consider the behavior of the commutator square in higherdimensional holographic theories, both in classical gravity and in string theory. Earlierdiscussion includes [4, 30]. For concreteness we consider the out-of-time-ordered correlator H ( t, (cid:126)x ) = (cid:104) W ( t, (cid:126)x ) V (0 , W ( t, (cid:126)x ) V (0 , (cid:105) (4.22)and the corresponding commutator C ( t, (cid:126)x ) = (cid:104) [ W ( t, (cid:126)x ) , V (0 , W ( t, (cid:126)x ) , V (0 , (cid:105) . (4.23)At the level of the gravity approximation, which corresponds to N → ∞ and λ h → ∞ inthe boundary theory , the exponential growth of H has the form [2, 4] H ( t, (cid:126)x ) = i c N e π/β ( t −| (cid:126)x | /v B ) (4.24)where c ∼ O (1) is real. The expression (4.24) comes from graviton exchanges which translatesinto the boundary theory from exchanges of the stress tensor. Such exchanges are fullycaptured by our EFT and we thus expect that in the classical gravity C = 0 , N → ∞ , λ h → ∞ (4.25)including the non-exponential parts.In Appendix A we show that within the eikonal approximation the sum of H and ˜ H vanishes identically in any theory dual to classical gravity. As such, there is no exponentialgrowth in C to leading order in 1 / N in classical gravity. In particular, we find that thiscancellation is a direct consequence of the fact the phase-shift for the bulk gravitationalscattering process is real. The suppression of the commutator OTOC in gravitational systemscan 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 ifthe bulk theory contains higher derivative terms reflecting certain finite coupling corrections. For example, when the boundary theory is the N = 4 Super-Yang-Mills theory, λ h is the ’t Hooft coupling.
16n order to find a non-vanishing expression for C at leading order in 1 / N , it is necessary toinclude not just the effects of gravitational scattering but also the exchange of stringy modesat finite string coupling α (cid:48) . This translates into the boundary theory as exchanges of aninfinite 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]. Thediscussion is a bit technical, however the qualitative features of these results are illuminating.The important result is that H 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 stringyregime for which | (cid:126)x | /t < v ∗ and a graviton dominated regime for | (cid:126)x | /t > v ∗ where for theblack hole background studied in [4, 30] we have v ∗ = d α (cid:48) v B R , v B = (cid:115) d d −
1) (4.26)with R the AdS curvature radius and d the number of spacetime dimensions in the boundarytheory.To elaborate, for short distances | (cid:126)x | < t/v ∗ , where v ∗ is some specific velocity, H canbe computed from an integral representation by a saddle point approximation and stringycorrections are important. In this regime it takes the form H ( t, (cid:126)x ) = c N t ( d − / f ( | (cid:126)x | /t ) e λt e − | (cid:126)x | Dt . (4.27)where f ( | (cid:126)x | /t ) is complex, the string corrected Lyapunov exponent λ is given by λ = 2 πβ (cid:18) − d ( d − α (cid:48) R (cid:19) . (4.28)and the constant D is given by D = d α (cid:48) β πR (4.29)In the regime | (cid:126)x | > t/v ∗ even at finite α (cid:48) , the integral for H is dominated by a gravitonpole. As a result the contribution from graviton exchange dominates, and H is given simplyby the classical gravity result (4.24). 17ere we point out that, for C (4.23), such a crossover does not exist, and its behavioris controlled by stringy exchanges for all t and (cid:126)x . This can be intuitively expected fromthat the gravitational contribution to C vanishes identically to leading order in 1 / N . Wegive technical details in Appendix A. There we show that the form of C at large N iscontrolled by the imaginary part of the phase shift for a bulk scattering, which is zero forgraviton exchange, but non-zero due to the effects of stringy modes. As a result, there isno graviton pole in the integral expression for C and there is no cross-over in its behavior.More explicitly, as shown in Appendix A, for all | (cid:126)x | /t , C has the form C ( t, (cid:126)x ) = c N t ( d − / f ( | (cid:126)x | /t ) e λt e − | (cid:126)x | Dt . (4.30)For | (cid:126)x | /t < v ∗ , there is a suppression in the prefactor compared with (4.27) (cid:12)(cid:12)(cid:12)(cid:12) f ( | (cid:126)x | /t ) f ( | (cid:126)x | /t ) (cid:12)(cid:12)(cid:12)(cid:12) = 2 sin (cid:18) πd ( d − α (cid:48) (1 − v /v ∗ )8 R (cid:19) , v = | (cid:126)x | /t (4.31)which can be through of as as a higher dimensional counterpart of the cos( λβ /
4) factor seen0 + 1 dimensional systems . V. A SIMPLE ARGUMENT FOR THE CROSSOVER VELOCITY
As we have just discussed, in addition to the above example of holographic systems atfinite α (cid:48) , 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 chaoticfor small v [16, 29]. A more elaborate discussion of this crossover, and further examples ofsuch systems, can be found in [30]. In this section we consider a simplest scenario for theexistence of such a crossover velocity .Consider higher spin exchange contribution to H defined in (4.22). At distances | (cid:126)x | (cid:29) β we expect the OTOC should decay exponentially with | (cid:126)x | . In other words, the time orderingshould not affect the validity of cluster decomposition principle at large distances. Denoting Note cos( λβ /
4) = sin( πd ( d − α (cid:48) / R ) and hence (4.31) reduces to 2 cos( λβ /
4) fior v = 0. We thank discussion with Gabor Sarosi and Mark Mezei for the content of this section. s operator to H as A s , we then expect A s ∝ e πβ ( s − t e − M ( s,β ) | (cid:126)x | (5.1)where M ( s, β ) is a function of spin s and the inverse temperature β . M ( s, β ) will alsodepend on any coupling constants in theory (e.g. α (cid:48) in string theory) and may also dependon other quantum numbers of the exchange operator. For example, for a two-dimensionalCFT in the large central charge limit, one finds that [3] M ( s, β ) = 2 πβ (∆ −
1) (5.2)where ∆ is the conformal dimension of the operator.Recall that the contribution of the stress tensor can be written as A T ∝ e πβ ( t − | (cid:126)x | ˜ vB ) (5.3)where ˜ v 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 | (cid:126)x | = vt if2 πβ ( s − − M ( s, β ) v ≥ πβ (cid:18) − v ˜ v B (cid:19) . (5.4)If (5.4) is satisfied for one s > s > v = 0, which is consistent with the fact that upon including finite-couplingcorrections 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 criticalvelocity v c such that for v > v c , equation (5.4) is not satisfied for any s >
2, while for v < v c it is satisfied for an infinite number of them. Then v c should provide an upper bound for v ∗ .Among all spin- s operators, let us denote M ( s, β ) as the smallest value for M . M ( s, β )can be considered as defining a finite temperature version of the “leading Regge trajectory.”Then existence of such a v c then implies that πβ ( s − M ( s, β ) − πβ v B ≤ v c , ∀ s > . (5.5)19n particular, the left hand side must have a finite limit as s → ∞ lim s →∞ πβ ( s − M ( s, β ) − πβ v B ≡ v ∞ ≤ v c (5.6) v ∞ can of course be larger or smaller than v ∗ . The above equation implies that M ( s, β )should increase with s at least as fast as linear dependence. In the case that v ∞ is nonzero, M must also be linear in s lim s →∞ M ( s, β ) = κ ( β ) s, v ∞ = 2 πβ κ ( β ) . (5.7)We stress that the above discussion is only a simplest scenario. It can happen that v ∗ existswithout existence of v c or v ∞ . VI. DISCUSSION
In this paper we have discussed various features of a maximally chaotic system, motivatedby the chaos effective field theory introduced in [21]. Clearly an important future questionis whether it is possible to write down an effective field theory for non-maximally chaoticsystems. Away from maximal chaos, an infinite number of operators contribute to OTOCs.Thus the key is whether there exists a finite number of effective fields which can capturecollective effects of the infinite number of operators. Right now there appears no obviousgeneral guiding principle which enables us to identify these effective degrees of freedom orwhat should govern their dynamics. Important clues may come from Rindler OTOCs in aconformal field theory where non-maximal chaotic behavior arises in the Regge limit fromresummation of contributions from infinite number of higher spin operators (see e.g. [30] andreferences therein).It is also of interest to better understand the physical interpretation of the vanishing ofthe leading order contribution to the commutator square for maximally chaotic systems.It seems likely this vanishing indicates a sharp distinction between the nature of operatorscrambling in maximally and non-maximally chaotic systems. One natural interpretation isthat this vanishing suggests an extra “unitarity” constraint on the scrambling processes of20aximally 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 precisein various contexts. In the gravity description given in Appendix A, the phase shift for thecorresponding string scattering is real in the maximally chaotic regime. Similarly, for RindlerOTOC, the vanishing of the commutator square means the vanishing of the imaginary part ofthe CFT scattering amplitude corresponding to the correlator [40]. These precise statementsresonate 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 inthe case of maximal chaos . Further, the existence of an enhanced “unitarity” for maximalchaos is related to the discussion of [29], where the coefficient of exponential growth incommutator OTOCs is inversely proportional to the branching time. As such this suggeststhat branching processes are absent in systems for which commutator squares vanish. ACKNOWLEDGEMENTS
We would like to thank Gabor Sarosi and Mark Mezei for helpful discussions. This workis supported by the Office of High Energy Physics of U.S. Department of Energy under grantContract Number DE-SC0012567.
Appendix A: Formula for commutator square in holographic systems
In this Appendix we present an expression for the commutator square in holographicsystems which is a simple generalization of the formula for OTOC in terms of the eikonalapproximation to the bulk scattering process presented in [4]. We will only briefly explainvarious notations below, and refer the reader to [4] for further details. We will consider a In [14, 31] this type of scrambling for maximal chaos has been referred to as being “coherent”. H ( { t , (cid:126)x } , { t + iθ, (cid:126)x } , { t , (cid:126)x } , { t + iθ, (cid:126)x } ) = (cid:104) W (cid:126)x ( t + iθ ) V (cid:126)x ( t + iθ ) W (cid:126)x ( t ) V (cid:126)x ( t ) (cid:105) ˜ H ( { t , (cid:126)x } , { t + iθ, (cid:126)x } , { t , (cid:126)x } , { t + iθ, (cid:126)x } ) = (cid:104) W (cid:126)x ( t + iθ ) V (cid:126)x ( t + iθ ) W (cid:126)x ( t ) V (cid:126)x ( t ) (cid:105) C θ ( { t , (cid:126)x } , { t , (cid:126)x } , { t , (cid:126)x } , { t , (cid:126)x } ) = (cid:104) [ W (cid:126)x ( t + iθ ) , V (cid:126)x ( t + iθ )][ W (cid:126)x ( t ) , V (cid:126)x ( t )] (cid:105) Recall that H defined above can be expressed as [4] H ( { t i , (cid:126)x i } ) = a (4 π ) (cid:90) e iδ ( s,b ) (cid:20) p u ψ ∗ ( p u , (cid:126)x ) ψ ( p u , (cid:126)x ) (cid:21)(cid:20) p v ψ ∗ ( p v , (cid:126)x (cid:48) ) ψ ( p v , (cid:126)x (cid:48) ) (cid:21) (A1)where the integral runs over (cid:126)x, (cid:126)x (cid:48) , p u , p v and the parameter a 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 functionof the centre of mass energy s = a p u p v and transverse separation b = | (cid:126)x − (cid:126)x (cid:48) | of the scatterredquanta. The various ψ i in (A1) are Fourier transforms of bulk-to-boundary propagators ψ ( p v , (cid:126)x ) = (cid:90) due ia p v u/ (cid:104) φ V ( u, v, (cid:126)x ) V (cid:126)x ( t ) (cid:105) (cid:12)(cid:12)(cid:12)(cid:12) v =0 ψ ( p v , (cid:126)x ) = (cid:90) due ia p v u/ (cid:104) φ V ( u, v, (cid:126)x ) V (cid:126)x ( t − iθ ) (cid:105) (cid:12)(cid:12)(cid:12)(cid:12) v =0 ψ ( p u , (cid:126)x ) = (cid:90) dve ia p u v/ (cid:104) φ W ( u, v, (cid:126)x ) W (cid:126)x ( t ) (cid:105) (cid:12)(cid:12)(cid:12)(cid:12) u =0 ψ ( p u , (cid:126)x ) = (cid:90) dve ia p u v/ (cid:104) φ W ( u, v, (cid:126)x ) W (cid:126)x ( t − iθ ) (cid:105) (cid:12)(cid:12)(cid:12)(cid:12) u =0 (A2)with φ V , φ W the dual bulk fields dual to Hermitian boundary operators V, W . Using theidentity˜ H ( { t , (cid:126)x } , { t + iθ, (cid:126)x } , { t , (cid:126)x } , { t + iθ, (cid:126)x } ) ∗ = H ( { t − iθ, (cid:126)x } , { t , (cid:126)x } , { t − iθ, (cid:126)x } , { t , (cid:126)x } )to compute ˜ H we find that the commutator C θ = H + ˜ H + . . . can be written as C θ ( { t i , (cid:126)x i } ) = a (4 π ) (cid:90) ( e iδ ( s,b ) + e − iδ ∗ ( s,b ) ) (cid:20) p u ψ ∗ ( p u , (cid:126)x ) ψ ( p u , (cid:126)x ) (cid:21)(cid:20) p v ψ ∗ ( p v , (cid:126)x (cid:48) ) ψ ( p v , (cid:126)x (cid:48) ) (cid:21) (A3) 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. Here we are only interested in the exponential growth of C θ , so are ignoring the time-ordered pieces δ ∼ G N ∼ / N , and hence an expansion in 1 / N amounts toexpanding the exponentials in (A1) and (A3). To leading order 1 / N we therefore find H ( { t i , x i } ) = ia (2 π ) (cid:90) δ ( s, b ) (cid:20) p u ψ ∗ ( p u , x ) ψ ( p u , x ) (cid:21)(cid:20) p v ψ ∗ ( p v , x (cid:48) ) ψ ( p v , x (cid:48) ) (cid:21) C θ ( { t i , (cid:126)x i } ) = − a (2 π ) (cid:90) Im δ ( s, b ) (cid:20) p u ψ ∗ ( p u , (cid:126)x ) ψ ( p u , (cid:126)x ) (cid:21)(cid:20) p v ψ ∗ ( p v , (cid:126)x (cid:48) ) ψ ( p v , (cid:126)x (cid:48) ) (cid:21) . (A4)For theories dual to classical gravity δ ( s, b ) is always purely real and can be computed byevaluating the on-shell gravitational action of a pair of gravitational shock waves [2, 4]. It isimmediately clear from (A4) that all C θ vanish identically to leading order in 1 / N in suchtheories within the eikonal approximation.In string theory the phase shift δ ( s, b ) obtains an imaginary part. For the black-holebackground discussed in [4] the phase shift δ ( s, b ) was found to take the form δ ( s, b ) ∼ G N s (cid:90) d d − k (2 π ) d − e i(cid:126)k · (cid:126)y (cid:126)k + µ ( e − iπ/ α (cid:48) s/ − α (cid:48) ( (cid:126)k + µ ) / r , s = c e πt/β (A5)where b = | (cid:126)y | , d − µ is a constant parameterrelated to the butterfly velocity, r is the horizon radius and α (cid:48) = l s (with l s the stringlength). We thus findIm( δ ( s, b )) ∼ G N s (cid:90) d d − k (2 π ) d − e i(cid:126)k · (cid:126)y (cid:126)k + µ (sin( πα (cid:48) ( (cid:126)k + µ ) / r )( α (cid:48) s/ − α (cid:48) ( (cid:126)k + µ ) / r . (A6)To see the behavior of (A6) let us first recall the evaluation of (A5) [4]. With (cid:126)y ≡ (cid:126)vt andthe explicit form of s plugged in, equation (A5) can be written as δ ( t, v ) ∼ G N c (cid:90) d d − k (2 π ) d − e i(cid:126)k · (cid:126)vt + πtβ (1 − B ( (cid:126)k )) (cid:126)k + µ ( c e − iπ/ α (cid:48) / − B ( (cid:126)k ) , B ( (cid:126)k ) = α (cid:48) ( (cid:126)k + µ ) / r , (A7)where v = | (cid:126)v | . At large t the integral in (A7) has a saddle-point at (cid:126)k ∗ where i(cid:126)v = 2 πβ ∂B∂(cid:126)k (cid:12)(cid:12)(cid:12)(cid:12) (cid:126)k ∗ = ⇒ (cid:126)k ∗ = i(cid:126)vµv ∗ , v ∗ ≡ πα (cid:48) µβ r . (A8)23he integrand in (A7) also has a pole at (cid:126)k + µ = 0 , µ = 2 πβ v B (A9)with v B the butterfly velocity. For v < v ∗ the integral can be computed using the saddle-pointapproximation. The resulting expression for the phase-shift δ ( t, v ) evaluates to δ ( t, v ) ∼ πDt ) ( d − / ˜ f ( v ) e λ ( v ) t (A10)where λ ( v ) is the velocity dependent Lyapunov exponent [29, 39] λ ( v ) = 2 πβ (cid:18) − d ( d − α (cid:48) R (cid:19) − v D , D = d α (cid:48) β πR (A11)and ˜ f ( v ) = G N c β d π ( d −
1) 1(1 − v /v ∗ ) ( c e − iπ/ α (cid:48) / − d ( d − α (cid:48) R (1 − v /v ∗ ) (A12)where in obtaining these expressions we have eliminated µ and r using that for the blackhole background studied in [4] we have v B = (cid:112) d/ d − r = 4 πRβ − d − with R the AdS radius. Recalling that the bulk to boundary wavefunctions in (A1) are spatiallypeaked around the positions of the external operators we have from (A4) that the functionaldependence of (4.22) is given at leading order in 1 / N by iδ ( t, v ) with v = | (cid:126)x | /t . This givesrise 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 existenceof a graviton pole at (cid:126)k + µ = 0 in the integrand of (A7). For v > v ∗ the contribution fromthe 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 (cid:48) ( (cid:126)k + µ ) / r )in the numerator. So in this case, the saddle-point approximation is valid for all (cid:126)x and t .We find Im( δ ( t, v )) = 1(4 πDt ) ( d − / ˜ f ( v ) e λ ( v ) t (A13)where now˜ f ( v ) = G N c β d π ( d −
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