Critical Quenches, OTOCs and Early-Time Chaos

In this article, we explore dynamical aspects of Out-of-Time-Order correlators (OTOCs) for critical quenches, in which an initial non-trivial state evolves with a CFT-Hamiltonian. At sufficiently large time, global critical quenches exhibit a universal thermal-behaviour in terms of low-point correlators. We demonstrate that, under such a quench, OTOCs demarcate chaotic CFTs from integrable CFTs by exhibiting a characteristic exponential Lyapunov growth for the former. Upon perturbatively introducing inhomogeneity to the global quench, we further argue and demonstrate with an example that, such a perturbation parameter can induce a parametrically large scrambling time, even for a CFT with an order one central charge. This feature may be relevant in designing measurement protocols for non-trivial OTOCs, in general. Both our global and inhomogeneous quench results bode well for an upper bound on the corresponding Lyapunov exponent, that may hold outside thermal equilibrium.


Introduction
Ergodicity is one of the main cornerstones of modern understanding of statistical mechanics and the dynamics of thermalization. While classical notion of ergodicity can be cleanly defined in terms of the phase-space of a system, in the quantum regime, this becomes subtle. The conventional measure of ergodicity or chaos in a quantum system is in terms of the energy-level statistics of the corresponding spectrum. In particular, level-repulsion is a tell tale sign of a quantum chaotic system [1].
On the other hand, given a quantum dynamics, various time-scales demarcate qualitatively distinct physics. The imprint of a level-repulsion appears on the real-time dynamics at the largest possible time-scale, the Heisenberg time t H ∼ ∆ −1 , where ∆ is the mean level-spacing and provides the smallest scale in the system. See e.g. [2] for a recent review on various time-scales in quantum dynamics. For a wide class of systems, quantum chaotic physics can appear at a much shorter time-scale, known as the scrambling time t * ∼ log N , where N is a parametrically large number, e.g. the number of degrees of freedom in the system. While the former can be associated with a late-time chaos, the latter is understood as an early-time chaos. In this article, we will focus on the latter.
Early-time chaos can be diagnosed in a special class of correlation functions, the Outof-Time-Order Correlators, OTOCs in short. Typically, an exponential piece in such a JHEP07(2022)046 4-point OTOC of a pair of operators translate into exponential growth in the expectation value of the commutator-squared of the corresponding pair. In the semi-classical limit, → 0, this intuitively parallels the exponential sensitivity of classical trajectories in the corresponding classical system. For thermal states with a temperature T , this behavior is visible above the dissipation time-scale t d ∼ T .
This class of correlators have recently found numerous new applications, ranging from the physics of disordered systems to the physics of black holes, see e.g. [3] for a review. Moreover, remarkable recent progress in quantum control of atoms and ions have picked an active interest in proposing measurement protocols for OTOCs (e.g. [4,5]) and early measurements on trapped-ion systems (e.g. [6]), which are closely related to the so-called Loschmidt echoes in spin systems and involve a quantum time-reversed evolution of the same. 1 While exciting progress is taking place in the experimental front, rather limited theoretical control is available on OTOCs in general, specially outside thermal states in systems with a large number of degrees of freedom. The best understood examples are in twodimensional conformal field theories (CFTs) with a large central charge [9], thermal states in CFTs with a Holographic dual in arbitrary dimensions [10], and some examples on thermal states in weakly-coupled large-N gauge theories [11,12]. All these examples warrant a thermal state and a large-N system. While thermal states are common, 2 experimentally, one would likely access a small-N system in which any non-trivial OTOC dynamics will only be transient.
In this article, we generalize both aspects. First, we move away from a typical state and consider a case when the state is not in an ab initio equilibrium. Secondly, we introduce a small parameter in the system, which makes it possible to have non transient chaotic behavior in some finite N examples, albeit in a perturbative regime. 3 We discuss these issues within the framework of the quantum quench protocol [13][14][15][16][17] , which provides a simple setup to study non equilibrium dynamics and thermalization in isolated quantum systems. The quantum quench dynamics, as the name suggests, refers to a process where the Hamiltonian of a system is changed over a very short time scale, for instance by a sudden tuning of some of its parameters. 4 One then studies time evolution of correlation functions of local operators and entanglement structure of the system following the quench. Interest in understanding the dynamics of quantum systems following a quench, has received an impetus since the seminal experiments involving ultracold atoms, where such a protocol was realized between superfluid and mott-insulating states [18,19].

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We will focus on systems, where the dynamics after the quench is a CFT. In [20], Cardy and Calabrese pioneered the study of the quench dynamics in these systems, and in particular computed the late time behavior of one point and two point functions of primary operators. The initial state of the system was chosen to be a very special state -the so called Cardy-Calebrese state(CC), e −τ 0 H |B , which is essentially a conformal boundary state (|B ) of the CFT [21,22], suitably regulated by e −τ 0 H , whose role is to cut off the high energy modes ( of order E > 1 τ 0 ) from the otherwise non-normalizable boundary state, thus making it a finite norm state. There are two fold advantages in choosing the CC state. Firstly, the computation of correlation function in the CC state is equivalent to a BCFT computation which makes the computation analytically tractable. Secondly, as argued in [20], the CC state may be physically thought of as approximating the ground state of a Hamiltonian which is an irrelevant deformation of the CFT. Moreover, the regulator parameter τ 0 maybe interpreted as the correlation length of this ground state. By computing one point and two point functions of primary operators in this state, the authors show that at late times, the results are identical to the ones in a thermal state, with the effective temperature being set by τ 0 . In effect, the CC state self-thermalizes at late times. This result is universal for all CFT's, so it holds for integrable as well as chaotic CFTs. This setup can be further generalized, for instance by introducing a spatial inhomogeneity in the original hamiltonian [23]. This could be done by adding impurities into the system which breaks the translational invariance of the system. This would effectively mean the correlation length of the ground state is now position dependent. In the critical quench scenario, this can be implemented by making τ 0 (x).
In this paper, we study the late time behavior of the OTOC of primary operators in the CC state as well as in a perturbed CC state in different examples of CFTs, including superintegrable, weakly integrable and chaotic CFTs. As examples of each we study the OTOC in a minimal model CFT, orbifold CFT and a large-c CFT respectively.
Firstly, we show that under a critical and homogeneous quench dynamics in (1 + 1)dimensions, in a large-c CFT 2 , the corresponding OTOC still exhibits a chaotic behavior similar to that of a thermal state -with an effective temperature set by τ 0 . Note that this result is non-trivial since it captures physics away from standard CFT-universality in two and three point functions. On the other hand, the effective thermal intuition ties naturally with the universal quench dynamics observed in the lower point correlations of [27], and selects out a class of 2D CFTs where this notion is much stronger. It is also interesting to place this result in light of a general effective thermal physics in 2D CFT with heavystates, see e.g. [28][29][30]. As a contrast, under a homogeneous quench, minimal models do not exhibit a thermal-like behavior. In this case, therefore, OTOCs can distinguish between an integrable system (e.g. the minimal models) and a chaotic system (e.g. large-c CFTs).
Secondly, we also demonstrate how an inhomogeneous critical quench, with a small inhomogeneity parameter that serves the role of an external field, can provide a scrambling window within which an exponential growth in OTOC can be distilled. Note that, it is not enough to have a small parameter in the system to warrant the hierarchy t * t d . For example, a weakly coupled field theory cannot do this, despite having a small coupling constant. On the contrary, to extract the Lyapunov growth of OTOCs, in the weakly JHEP07(2022)046 coupled regime, one needs to carry out a perturbation series in high-orders of the coupling constant and subsequently resum them. Thus, the final result becomes, in a certain sense, non-perturbative in the coupling constant. See e.g. [11,12] where such weakly coupled results are obtained, necessarily with a large-N system.
Our results show that the small inhomogeneity parameter a, which is supplied from outside, can indeed provide us with the hierarchy. In fact, we show with an example that an integrable 2D CFT with c = 2 as well as the minimal models, subject to an inhomogeneous quench, indeed develops a chaotic exponential growth supported by this small parameter. Subsequently, the scrambling time can be separated from the scale τ 0 by a factor of (− log a). In these cases, the deformed CC-states can be thought of as the atypical states for which a thermal-analogue result holds, irrespective of whether the Hamiltonian is integrable or not.
This article is divided into the following parts: in section 2 we give a brief review of the quench set up in CFT 2 [27]. Section 3 and 4 contains the bulk of our analysis and results. We consider two complimentary limits in the paper. In section 3 we study the limit τ 0 → 0. After setting up the basic framework in section 3.1, we study a special 3-pt OTOC in the CC state, for the homogenous quench in a large-c CFT in section 3.2. We explicitly show that at late times, the OTOC shows a maximal Lyapunov growth with an effective temperature β = 4τ 0 . We then analyze the same 3-pt OTOC for a large-c CFT after introducing a small inhomogeneity in the quench setup in section 3.3. We show, that at leading order in the perturbation, the Lyapunov exponent remains the same, though there is a change in the butterfly velocity. In section 4, we focus on bulk 4-point correlators in the limit τ 0 (t, x). We study three examples-the large-c CFT (section 4.1), minimal model CFT (section 4.2) and an orbifold CFT (section 4.3). For the case of the large-c CFT, we show that in the presence of the inhomogenous perturbation, the effective Lyapunov exponent changes due to the perturbation. In the case of the minimal model CFT, and for the orbifold CFT, the perturbation induces an exponentially growing piece in the corresponding OTOC, and thus exhibits a chaotic behavior. Moreover, the small inhomogeneity parameter induces a large hierarchy between τ 0 , and the scrambling time. We end with a summary of the key conclusions of our paper and a discussion of some future directions in section 5.

Correlation functions: general set up
Vacuum correlation functions in a Lorentzian QFT are typically defined by an analytic continuation in the time coordinate via the i prescription. (t → τ = t − i ).
Where τ ij = τ i − τ j and the Hamiltonian H is bounded from below (assumed here for simplicity to be a positive semi-definite operator). The e −H ij factor provides a UV regulator as long as ij > 0. Thus the r.h.s. is well-defined provided ( 1 > 2 > · · · > n ). This procedure however fails if we compute correlation functions in an arbitrary state (say |Ψ 0 ), which is not an eigenstate of the Hamiltonian, since by the same argument as given JHEP07(2022)046 above, is analytic only when (0 > 1 > 2 > · · · > n > 0 ), which is impossible to satisfy. One way to get around this and get a finite analytic region is to introduce a regulator (τ 0 ) which effectively cuts off the very high energy modes with energy greater than 1 τ 0 . This may be achieved by substituting the state by the regulated state |Ψ 0 → e −τ 0 H |Ψ 0 . Here τ 0 is necessarily non-zero positive real number. In this regulated state, the correlation function maybe defined via an analytic continuation as above, provided we restrict (τ 0 > 1 > 2 > · · · > n > −τ 0 ). This essentially means that the Euclidean theory whose analytically continued version is described by the real time correlation function is defined on a strip-ie: . Thus in the Euclidean picture, the correlation function computation becomes a computation on a strip of length 2τ 0 , with the boundary conditions at the ends of the strip being determined by the state |Ψ 0 . Note that, this picture is also valid for evaluating Out-of-Time-Order Correlators (OTOCs), for which an ordered n-tuple of { i } decides the corresponding operator (time)ordering in the correlator. 5 In applications to quantum quench problems, the state |Ψ 0 is taken to be an eigenstate (typically the ground state) of the original Hamiltonian (H 0 ) before quenching. In [3], Cardy and Calabrese studied a quantum quench from a massive Hamiltonian H 0 to a CFT Hamiltonian H. If H 0 is close to the CFT Hamiltonian H in the RG sense, ie H 0 is obtained from H by a small irrelevant deformation, then [3] argued that correlation functions in such a state over time scales and length scales much larger than the correlation length (ξ) of the state, show universal behavior which may be captured by replacing the state by a regulated conformal boundary state in the fixed point CFT.
Thus in this approximation, we may take the state |Ψ 0 to be the conformal boundary state in the BCFT on a strip. The regulator τ 0 now has a physical interpretation as the correlation length of the state in consideration ie: (τ 0 ∼ ξ). Thus the Euclidean correlation function calculation reduces to a BCFT computation on a strip.
This set-up may be generalized to incorporate the case when the state in consideration has a spatially inhomogeneous profile (ie ξ = ξ(x)). Atleast when the inhomogeneity is small, this can be modeled by replacing the constant width Euclidean strip 2τ 0 by a strip with a position dependent width 2τ 0 (x).
In CFT, the 'variable width strip' geometry can be mapped, via a suitably chosen conformal transformation, to a 'constant width strip' geometry, thus reducing the problem to the homogenous quench. Following the notation of the original work, we refer to the 'variable width strip' geometry as (VWS) and the 'constant width strip' as (CWS). Let the coordinate on VWS be ω = x + iτ where at any value of x, τ ranges from −τ 0 (x) to τ 0 (x) and that of CWS be ζ =x + iτ where −τ 0 <τ < τ 0 . Then the conformal map from JHEP07(2022)046 VWS to CWS, g : ω → ζ = g(ω), must satisfy the boundary condition For the simplified case where the inhomogeneity is taken to be a small fluctuation over τ 0 , an explicit solution for the map g(ω), or more precisely its inverse has been obtained in [23]. Parameterizing the infinitesimal map as g −1 (ζ) ≡ ζ + f (ζ), one finds that the inverse map must satisfy the following boundary condition: 6 Imf (x ± iτ 0 ) = ±h(x) .
If one further assumes that the VWS geometry has a reflection symmetry about τ = 0, 7 then f (ζ) can be determined, upto an irrelevant real constant, in terms of the function h(x) as: Here h(x) has been normalized to satisfy h(−∞) = 0. An interesting limit, which is useful in this quench scenario, is taking τ 0 → 0. In this limit, 2.4 reduces to the following Correlation functions of primary operators in the VWS geometry is related to that of the CWS geometry in the standard way.
(2.6) In the next section, we will study the OTOC in the inhomogeneous quench set up in two extreme limits τ 0 → 0 and the opposite large τ 0 limit.

The basic framework
The first case we study is a bulk-boundary three point OTOC function on the VWS. It was demonstrated in [31] that this class of 3-point correlator capture OTOC-dynamics and from hereon we refer to them as b-OTOC. The scalar operator W of dimension h W (=h W ) is placed at (x, 0) and two boundary scalar operators V of dimension h V are placed at the two boundaries (0, i(τ 0 + h(0))), (0, −i(τ 0 + h(0))) on the VWS. We want to compute Euclidean correlator of the type V (ω 1 ,ω 1 )W (ω 2 ,ω 2 )V (ω 3 ,ω 3 ) . In the current set up, (ω i ,ω i ) i=1,2,3 are the following:

W < l a t e x i t s h a 1 _ b a s e 6 4 = " i / N c n x 9 g H j F O y 6 C M H i T n R J F t k P w = " > A A
< l a t e x i t s h a 1 _ b a s e 6 4 = " 3 e 0 W y D 5 J 1 N l p 9 y Q j r 7  Whileω i take the corresponding complex conjugate values. We will then finally analytically continue to real time with (ω 1 → t + i(τ 0 + h(0)), ω 2 = x and ω 3 → t − i(τ 0 + h(0))). We will be interested in the large real t limit and the observable we would like to compute is the normalized correlation function V W V V V W . We calculate it by first mapping this to the CWS geometry and then further mapping it to the UHP geometry. On the UHP we use the doubling trick to compute the correlation functions. This whole set up is pictorially described in figure 1.
Using the map from VWS to CWS, we obtain: Correlation functions on the CWS geometry can be mapped on to the UHP geometry by the standard map (z = −ie π 2τ 0 ζ ). Thus the boundary points ζ 1 , ζ 2 on the CWS are mapped to the points z 1 , z 2 on the real line. As is well known, this bulk boundary three point function, has the same structure as that of a holomorphic four-point function in the full plane and so the normalized three point function on the CWS is of the form

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One can now compute the VWS correlation functions at the point ω i by expanding ζ i upto first order in f expansion, (3.5) Here in the second line, we have used the boundary condition of g(ω) as in (2.2) to write F (η(ω i ,ω i )) as a CWS bulk-boundary three point correlator where two boundary operators are located on the boundary of the strip ±iτ 0 . After explicitly using (3.1) and the reflection symmetric property of f , we finally get (3.6) Hence the central object of the three point bulk-boundary OTOC computation in this inhomogeneous quench process, is the quantity F (η(ω i ,ω i )). In other words, F determines the three point OTOC for global or homogeneous quench which is described purely by OTOC in CWS. The amount of inhomogeneity in the correlators at VWS is injected through the f (x). As we see from (3.6), in this special bulk-boundary setting, only spatial inhomogeneity enters in the first order correction.
Based on (3.6), we can already anticipate the effect of inhomogeneity. For example, suppose F (x, t) ∼ e λ(t−x/v B ) , then inhomogeneity affects the butterfly velocity such that: Let us discuss the implications of the above result. First, on physical grounds, it is natural to expect that the butterfly velocity provides us with the maximum speed at which information can propagate in the system, 8 and therefore for any state, v B ≤ 1. Thus, while for f (x)/x < 0, this is trivially satisfied, it is not so when f (x)/x > 0. For the latter, when the homogeneous quench yields v B = 1, v B > 1 and therefore seems to violate a causality bound. 9 On the other hand, a priori there is no constraint on the sign of f (x) and therefore can cause an apparent violation of causality. Let us carefully explore the logical possibilities. First, one compelling possibility is that this violation is a consequence of the leading order perturbative analyses and may be restored when higher order effects in f (x) are included and perhaps resummed. In this case, the specific value of the butterfly velocity may not be trustworthy but the existence 8 Note that the Lieb-Robinson bound, for a system, is state-independent as it is defined in terms of the L∞-norm of operator commutators. 9 Recall that, even though the initial state is not Lorentz-invariant, it is nevertheless evolving with a CFT Hamiltonian after the quench. At sufficiently large times, the information propagation should be constrained by how fast information can propagate with the CFT-Hamiltonian. Thus, for a Lorentz-invariant CFT, we expect that the butterfly velocity should be upper bounded by the speed of light.

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of such a velocity is still consistent. The upshot is that the existence of a chaotic feature can be inferred from the perturbative result. Secondly, as has also been noted in [45], although it is natural to expect that in a relativistic system the butterfly velocity will obey the causality constraint, the region that needs to be measured to reconstruct a local perturbation can grow faster than the speed of light. This will cause the butterfly velocity to violate the speed of light, although no physical violation of causality takes place. In particular, the superluminal butterfly velocity in [45] exists outside any perturbative framework.
Finally, it is of course possible that the butterfly velocity exceeds the speed of light in a theory with an explicit breaking of Lorentz-symmetry, see e.g. [46,47]. From the discussion and examples above, it appears that the butterfly velocity can violate the causality bound when no such violation of Lorentz-symmetry is taking place. The essential take-away message for our purpose is only that we trust the existence of a butterfly velocity (and the Lyapunov exponent) when we see one and therefore we trust the existence of a chaotic behavior in such a scenario.
Later, we will discuss explicit examples in which this possibility is further evidenced. Let us now discuss the homogeneous quench case in more details.

A non-trivial 3-point b-OTOC: homogeneous quench
The homogeneous quench limit can be easily obtained by setting f (x) = 0, in the discussion above. This yields, using equation (3.6), the desired 3-point correlator in terms of the function F (η). Before discussing the 3-point b-OTOC, let us begin with lower point correlators.
The basic features are nicely summarized in [27], which we will heavily draw on in the subsequent discussion. First, an n-point function of primary, scalar operators, denoted by Φ i (ω i ), on the CWS can be mapped to the UHP. Recall that ω = x + iτ . Subsequently, to obtain the real-time correlator, we analytically continue: τ → τ 0 + it and take the limit {t, x ij } τ 0 to obtain the asymptotic time-dependence. It is straightforward to obtain the large time behavior of a one-point function of a scalar, primary, of dimension ∆ Φ : Φ(t) ≈ (π/2τ 0 ) ∆ Φ e −∆ Φ πt/(2τ 0 ) . It defines a relaxation time for the corresponding operator: t rel ∼ 2τ 0 /(∆ Φ π). This already indicates that τ 0 sets a dissipation-scale in the dynamics.
A more direct and explicit understanding is given by the one-point function of the stress-tensor. On the UHP, T µν UHP = 0 and therefore on the CWS, the sole contribution comes from the Schwarzian piece of the conformal transformation. This yields: where c is the central charge. Comparing this with a thermal one-point function, we can read-off an effective temperature β eff = 4τ 0 . Similarly, twopoint correlators can be also explicitly obtained in the asymptotic time limit. Using the results in [24], it is easy to see that the connected equal-time correlator between two scalar primary operators, in this limit, is given by:

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at the leading order. 10 The behavior in (3.8) hints at the existence of an approximate light-cone at t = /2. This effective light-cone is further supported by the dynamical behavior of entanglement entropy of a line-segment (denoted by A) of length [20]: that clearly demarcates a linearly growing behavior in time, from a plateaux. This behavior can be easily understood from a ballistic propagation of the quasi-particles produced by the initial state, with a propagation velocity v B = 1. Dynamical behaviors in both (3.8) and (3.10) are identical to that of a thermal state of temperature β eff = 4τ 0 . Furthermore, as is explicitly shown in [20], the reduced density matrix of a sub-system of length , after sufficiently long time, becomes exponentially close to a thermal density matrix, with the same β eff . The arguments above generalize for an n-point function. The lesson, therefore, is that at the large time limit, global (homogeneous) critical quench dynamics can be described by an effective thermal physics.
Let us now consider the 3-point b-OTOC in detail. Here, g(ω) = ω in figure 1, we only need the map of the CWS to the UHP. On the CWS, the operators are placed at V (+τ 0 , 0), V (−τ 0 , 0), W (0, x). As is already mentioned in the previous section, we will place the operators at ω 1,3 = ±iτ 0 and ω 2 = x, and subsequently analytically continue the result to ω 1 → t + iτ 0 and ω 3 → t − iτ 0 . Keeping track of this analytic continuation, the points ω 1,2,3 are mapped to z 1 = −ie b(t+i 1 ) , z 3 = −ie b(t+i 2 ) and z 2 = −ie b(x+i 0 ) , with b = π/(2τ 0 ). Note that, we have included i 0 in the location of the W operator, which will be used to define the time-ordering. Furthermore, 1 = τ 0 and 2 = −τ 0 ensures that, at t = 0, z 1,3 are points on the Im(z) = 0 boundary of the UHP. 11 Within this set-up, there exists only one relevant -ordering i.e. 1 > 0 > 2 , for which one obtains an Out-of-Time-Ordered Correlator (OTOC).
With these assignments, the invariant cross-ratio in (3.3) can be calculated, the details are given in equation (A.2). As described in equation (A.3), in the limit t → 0 as well as t x, η → 0 from opposite directions, but for the OTOC-configuration, i.e. when 1 > 0 > 2 , η can be larger than one. In the complex η-plane, one therefore moves to the second Riemann sheet, while crossing a branch-cut running from unity to infinity. This is the essential kinematic aspect that distinguishes between a TOC and an OTOC [9]. The dynamical information is contained further in the function F (η), for a given CFT. 10 The correction is of the order O(e − π ∆ Φ τ 0 ), which is exponentially suppressed as compared to e − π ∆ Φ 2τ 0 (a term that is itself small in the τ0 limit). 11 In principle, we can keep 1,2 independent of τ0. However, in doing so, will map the V operators away from the UHP-boundary. This is also an interesting correlator, however, since the V operators move away from the UHP-boundary, they will generically be given by a six-point holomorphic function in the entire complex-plane. Constraining the V operators on the UHP-boundary provides us with the simplest non-trivial OTOC for the system [31].

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An interesting class of CFTs in which this can be explicitly calculated is the large c CFTs, presumably with a Holographic dual. In this case, one takes the limit that h v is fixed and large while h w /c is fixed and small, 12 in which case the explicit result is analytically known [44]. 13 The complete answer for the 3-point b-OTOC takes the following form: (3.13) where, the scrambling time t * = β eff /(2π) log c and 12 = i(e i 1 − e i 2 ). Clearly, t * is parametrically large compared to the dissipation scale, set by β eff . The corresponding exponential growth of the OTOC are characterized by a "maximal" Lyapunov λ L = (2π)/β eff 14 and a butterfly velocity v B = 1. The light-cone observed in (3.8) matches exactly with this velocity. Finally, note that the result in (3.12) holds for any finite 12 , which works as a regulator and cuts off arbitrarily high energy modes. Equivalently, one can smear the operators V over a Lorentzian time-scale ∆τ , such that each infinitesimal 12 → ∆τ , in the formulae above [9]. Thus, all conclusions above hold for any finite value of τ 0 .
The discussion of this section is consistent with, and reinforces, the observation made in [20], that the CC state at late times behaves as a thermal state. Given this fact, it is natural to wonder whether some version of the chaos bound derived in [10], exists for OTOC's in the CC state.
The key mathematical content of the proof, presented in [10], is the existence of an upper bound on the rate of growth of the regulated four-point function (F ); in particular an upper bound on 1 1−F (t) | dF (t) dt |, given in equation (4.1) of that paper. As shown in that paper, such a bound exists when the following criteria are satisfied: (1) the function F (t + iτ ) is analytic inside a semi infinite strip t > 0 and − β 4 ≤ τ ≤ β 4 , (2) F (t+iτ ) is real for τ = 0 and Coming back to the CC state, we expect much of the above argument would go through, provided we choose for F (t) = <V (0−iτ 0 )W (t)V (0+iτ 0 )>

<V V ><W >+
, where the correlation functions are computed in the BCFT. By the doubling trick, this would behave like a four-point function in a bulk CFT, and so we expect that arguments similar to what went into showing that condition (3) is met would still hold in this case. In particular at late times when the CC state behaves like the thermal state, the factorization property of the correlation function would also hold. And so, as before with a suitably defined , the corresponding F satisfies condition(3). Condition (1) and (2) is again satisfied by our choice of F . The bound should then follow along the same lines as given in section 4 of [10].

2-point correlation function.
Let us begin with a discussion on the 2-point correlator, as we did for the homogeneous quench. The disconnected correlation function can be calculated easily, see e.g. [23]. Without any loss of generality, let us discuss a specific The equal time two point correlation function in VWS, in the limit τ 0 → 0, is given by: In the second line of the above equation we have considered be π(x i −t) 2τ 0 1, for i = 1, 2. Thus for, t x 2 , the above equation yields: Therefore, due to the sub-leading contribution, there is a growth in 't' at late times.

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Therefore, for t x 1 , x 2 , there is an overall decay at early times, even after considering sub-leading contributions. Using these, we calculate the connected correlation function φφ − ( φ ) 2 . This yields a vanishing answer for t < (x 1 −x 2 ) 2 . Hence, the equal-time 2point function sees the same light-cone as it does for the homogeneous quench. Perhaps more importantly, the light-cone has no memory of the initial inhomogeneous state and coincides with the relativistic CFT light-cone. We will see momentarily that higher point functions, specially the OTOCs encode a different light-cone altogether.

Higher-point functions & OTOCs.
We will now discuss specific examples, in which we compute the 3-point b-OTOC in explicit details. The simplest example is large-c CFTs, in which, as before, we take the W operators to be heavy and the V operator to be light. Using the explicit result for the identity block in equation (A.7), and substituting in (3.5), one obtains: where t * = (2τ 0 /π) log c and we have kept only leading order terms in an (1/c)-expansion.
Below the scrambling time, the corrections due to inhomogeneity can be exponentiated 15 to yield: 20) From the second line above, we can read off the corresponding Lyapunov exponent as well as the butterfly velocity: λ L = π/(2τ 0 ) and v B = 1 + (2a)/τ 0 , for τ 0 a. While λ L receives no correction, there is a change in the light-cone structure since v B is modified. Correspondingly, the scrambling time is modified in (3.20), with an order one change: t * ∼ (1/λ L ) log ce −aπ/τ 2 0 b . As we now explicitly observe: v B > 1 when a > 0. To better understand this, consider the example: h(x) = aΘ(x − a 1 ), with a τ 0 and a 1 is arbitrary. This yields: f (x) = a(x − a 1 )Θ(x − a 1 ). Repeating the same calculation as JHEP07(2022)046 above, we get when t > t * . (3.22) Re-arranging the above expressions in (3.22), we can read-off the Lyapunov exponent and the butterfly velocities as: ) with a τ 0 and a 2 > a 1 . The Lyapunov exponent remains unchanged, as above. The butterfly velocity, on the other hand, is given by It is natural to wonder -since the inhomogeneity is introduced by a simple step function -why the Lyapunov exponent only sees τ 0 and not (τ 0 + a). Indeed, physically, λ L = π 2(τ 0 +a) in the limit a 1 → −∞. It is easy to check, however, that in a perturbative expansion in (a/τ 0 ), the shift in τ 0 → τ 0 + a in the expression of λ L = π/(τ 0 + a) is only visible after summing over all the higher order corrections starting from O(a/τ 2 0 ). In the same limit, we also expect v B = 1, since this is arbitrarily close to a homogeneous quench.
This simple exercise suggests that the superluminal butterfly velocity we observe in this example has its origin in the perturbative treatment of our analysis and that an exact treatment incorporating all higher order corrections in O( a τ 2 0 ) would yield an expression for the butterfly velocity consistent with causality. However, we repeat that even though the value of the butterfly velocity is not trustworthy in this case, its existence certainly is.

A 4-pt bulk OTOC: in τ 0 x, t limit
In the previous section, we have seen that a three point b-OTOC already captures a change in the light-cone structure and in the scrambling time, at the leading order perturbative calculation in inhomogeneity, for a large-c critical quench. It is further clear from equation (3.6) that the temporal part remains unaltered in the 3-point b-OTOC. We will now show, by analyzing higher point functions, that the Lyapunov exponent also receives corrections due to inhomogeneity of the initial state. Our arguments will rely on general structure of higher point functions in the limit where bulk operators approach close to each other, compared to their proximity to the boundary. To see this effect, we consider two scalar operators V of dimension h V (=h V ), placed at (0, iτ ) and two other scalar operators W of dimension h W , placed at (x, 0) on the VWS and follow the same map as before i.e to first map operators from VWS to CWS and then to UHP. In UHP, the probe operators are far away from the boundary if τ 0 x, t, where t refers to the analytical continuation of τ . 16

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We consider, the four-point correlator of the type V (ω 1 ,ω 1 )W (ω 2 ,ω 2 )V (ω 3 ,ω 3 )W (ω 4 ,ω 4 ) where, (4.1) andω i → corresponding complex conjugates and compute the normalized correlation func- The bulk four-point function on the UHP has the same structure as a holomorphic eight point function in the full plane. However, in the limit τ 0 x, t, we can ignore the boundary effect. This implies that in the UHP, the eight point function is factorized into a product of two 4-point functions of bulk and it's image points i.e F (z i ,z i ) ≈ g(z i )g(z i ). 17 Here z i = ie πω i 2τ 0 , as we have mentioned earlier. Notice that this map is exactly same as the vacuum to thermal (or plane(ω) to cylinder(z)) map with temperature β i.e. ω → z = e 2πz β [20]. 18 Hence, in this small τ 0 limit, the decoupled fourpoint OTOCs give the same answer as that in a thermal ensemble with the identification β = 4τ 0 , adding further evidence to our previous discussion based on the 3-point b-OTOC. This τ 0 (x, t) limit is tractable in both homogeneous as well as inhomogeneous quench. Proceeding as before, we find the normalised correlator on VWS: After carrying out the analytic continuation using i prescription from iτ → t + i , we get: which has manifest temporal derivatives and therefore will affect the time-dependence.
Here we choose 2 = 4 = 0, while 1 and 3 are taken to be positive. We then have the following ordering: ( 1 > 2 < 3 > 4 ), which in an OTO configuration as discussed in section 2.
In the limit τ 0 . As we argued before, this g can be computed from the thermal four-point OTOC in CFT 2 . Hence, the correction due to inhomogeneity comes purely from the temporal and spatial derivatives in (4.3).
As before, based on (4.3) we can already anticipate the effect of inhomogeneity on the corresponding Lyapunov exponent. First of all, for an monotonically increasing function JHEP07(2022)046 f (x)(andf (t)) taking a schematic functional dependence F (x, t) ∼ e λ L (t−x/v B ) , one can have x, f (t) t, we obtain: As before, the correction to butterfly velocity can exceed speed of light by an amount 4f (x) . Also it is easy to observe that for an even function f there is no correction to λ L and v B = v B 1 + 4f (x) x . However for an odd function, f . Even in this case for t x, v B > v B . As we have argued before, this violation is likely to be visible only in the leading order perturbative answer. We emphasize again that, within the most conservative interpretation of this observation, we infer the existence of a Lyapunov exponent and a butterfly velocity. Of course, the theory was chaotic before the perturbation and remains so after it.
More generally, from (VWS final 4pt), it is clear that for any system that has a nontrivial time-dependence in the CWS correlator, the corresponding VWS correlator will have a universal exponential growth, provided by the external data f (t) ∼ e κt , with a Lyapunov exponent set by κ. Moreover, the term denotes the maximum modulus of the function f (t), which by assumption is still a small number and τ 0 appears on dimensional grounds. Therefore, we obtain a natural definition of scrambling time, without making any reference to large number of degrees of freedom. Later we will discuss an explicit example of this. More generally, it is clear from the analyses leading to (4.3) that higher point functions will also have a similar behavior.
To further elucidate with explicit examples, we use (2.4) to obtain f (x) in this τ 0 limit. For analytical control, we choose: h(x) = a e πx/τ 0 1+de πx/τ 0 , such that (a/d) τ 0 . This yields: Furthermore, the e 2πz/τ 0 -term in the above expression can be ignored by arranging sufficiently small d. With this, let us now discuss specific examples.

Large c CFT 2 :
This is an interesting class of CFTs, for which the 4-point CWS-correlator is given by a thermal correlator with an effective β eff = 4τ 0 . Using the identity block dominance, the corresponding thermal correlator is obtained to be:

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For a marginal operator, however, there is no time-dependence and for an irrelevant operator, the sub-leading piece decays exponentially. Note that, although this is an appealing observation, we cannot make a general claim about this structure since our analyses above depends explicitly on the choice of the inhomogeneity function h(x). It will be interesting to explore a potentially underlying structure in future.
Before moving on further, let us contrast the above results with a free CFT. For these, the late time OTOC is dictated by a constant phase factor [38]. This can be argued simply due to factorization of four-point OTOC into product of two point functions which differ from TOCs by phase factors. That is for free fields, the Euclidean W W V V → W W V V and due to i prescription a phase factor appears when we divide the OTOC by a TOC, after analytic continuation in real time. Hence, inhomogeneous quench does not induce any non-trivial time-dependence for such free theories. Thus integrable free systems remain integrable.
It is worth comparing the above examples with what happens in classical chaos. For classical integrable systems with M degrees of freedom, the phase space is foliated by an M -dimensional tori. Here, the number of periodic directions, M , corresponds to the number of conserved charges. The celebrated KAM-theorem states that small, non-linear perturbations do not destroy this tori structure and therefore classical integrability is retained in the perturbative limit. For free CFTs, the OTOCs display a similar structure, for small perturbations. Thus, while the atypical CC-state or its appropriate deformation can behave like a thermal state, as far as the OTOCs are concerned, for free theories the OTOCs are trivial. Therefore, even with these atypical states, OTOCs can distinguish between an interacting integrable system from a free integrable system.

Orbifold CFT
Correlation function of twist operators and its growth has been extensively studied in [40] to understand growth of Renyi entanglement entropy for subsystems in QFTs. In the cyclic orbifold CFT, these twist operators exist as natural candidates of primary operators. In [41], OTOCs for twist operators in the orbifold CFTs has been studied as an attempt to characterize different CFTs in terms of chaotic properties alongside entanglement growth. In this section, we will use these results to study the change of OTOC due to inhomogeneous quench.
In concrete term, we consider free c = 2 CFT on cyclic orbifold (T 2 ) n /Z n , where T 2 = S 1 × S 1 (or, two free bosons are compactified on the same radius R). Furthermore, let us take W and V as twist operators σ n with dimension h n =h n = 1 12 n − 1 n . The corresponding OTOC depends on the compactification parameter η = R 2 . In the limit JHEP07(2022)046

Conclusions
Our conclusions can be summarized into two main proposals. First, in critical quench dynamics, the universal effective thermal physics of the lower point functions [27] do generalize for a larger class of correlators, including the OTOCs for chaotic CFTs. However, for a homogeneous quench, OTOCs also distinguish a chaotic CFT from an integrable CFT. While we have used large-c CFTs for the former and minimal models for the latter, as our examples, we expect these features to survive for a much wider range of systems. Our expectation is motivated from how CFT predictions qualitatively hold in a wider class of quench systems, see e.g. [27] for some explicit examples.
From a Holographic perspective, the non-equilibrium dynamics of the critical homogeneous quench appears to be well-approximated by a suitable black hole geometry. Intuitively, this is similar to setting up a non-trivial boundary condition, of some field, at the AdS-boundary, and letting it evolve in time. If one now explores correlators at large time limit, due to Birkhoff's theorem, the dynamical geometry, away from the bulk fields, will be given by the Schwarzschild metric. This simple intuition transcends dimensions and hence it is suggestive of a broader applicability of the effective thermal description in dynamics. 21 Also, the basic assumptions [10] of cluster decomposition and analyticity are expected to hold for homogeneous quench as well. Taken together, our first proposal/conjecture is: for chaotic systems, under homogeneous quench, the corresponding Lyapunov exponent is upper bounded by the mass scale associated to the initial state. This conjecture is further consistent with numerical results in [32], which investigates a non-critical quench with Ising spins. Inhomogeneity provides some perturbative corrections to this physical picture. It will be an extremely interesting problem to prove a bound for these cases in general.
Our second proposal is related to how an integrable system, under a suitable deformation, becomes chaotic. While this is not surprising, explicit examples of such, in the realm of early-time chaos are hitherto unavailable. In this article, we explicitly demonstrated, using the minimal models and the integrable orbifold CFT, how an inhomogeneous quench leads to chaotic dynamics in these systems. Moreover, the small inhomogeneity parameter defines a large scrambling time-scale that allows one to access this chaotic behavior. While the example is specific, corresponding technical aspects are rather generic and are expected to hold for a wide class of CFTs with an order one central charge.
This CFT can furthermore be perturbed by a relevant operator, within a conformal perturbation theory. The corresponding OTOC will receive an additional contribution coming from a higher point function integrated over the entire complex plane. This correction, generally, will not cancel the inhomogeneity-induced exponential growth. Motivated by this, we propose/conjecture that non-trivial OTOC-dynamics for a generic small-N system can indeed be detected with an inhomogeneous quench set up. It would certainly be very interesting to explore this in a specific model in future, specially in the context of recent advances in experimental protocols of OTOC measurements, e.g. in [42].

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From the perspective of quantum properties of black holes, it would be particularly interesting to understand how a Holographic calculation restores v B = 1 for a general inhomogeneous quench. On the other hand, real-time dynamics of the dual CFT often encodes physics behind the horizon of the black hole, see e.g. [43] that captures universal features of time-evolution of entanglement entropy. Inhomogeneous quench is particularly interesting from this perspective, since a corresponding boundary data will likely affect the singularity structure inside the black hole. These will be very exciting future directions to further uncover.

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In this case, there exist two inequivalent time-orderings i.e. 0 > 1 > 2 (time-ordered) and 1 > 0 > 2 (out-of-time-ordered). The ratio 02 10 is < 0 for time ordering and > 0 for out of time ordering. Thus η| t=x > 1 for OTOC and < 1 for TOC. This distinct behavior of η plays a crucial role in determining the late time behavior of the correlators, as we will review now.
The Euclidean correlator we consider now is the following: In large c limit, with hw c is fixed and small while keeping h v large and fixed Virasoro vacuum identity block F (η) can be obtained analytically [44]. Assuming further an identity block dominance, the full conformal four-point function f (η) can be approximated by the identity block contribution. The analytic expression of this block contains branch point at η = 1 with the cut [1, ∞). Following a contour around η = 1 branch point and taking the small η limit, one obtains [9]: 23 (A.5) Now we substitute the large time behavior of η. Define Σ ij = i(e iα i − e iα j ) and Σ * ij = −i(e −iα i − e −iα j ), the large time limit of η becomes: Using this in (A.5), we get: Where the scrambling time t * = 1 α log(c). Thus, we obtain the corresponding Lyapunov exponent: Comparing with the usual maximal Lyapunov exponent at large c i.e. λ L = 2π β , we obtain an effective inverse-temperature is β = 4τ 0 . This is consistent with the effective temperature in homogeneous quench case as in the literature [23].

B Four-point OTOC at CWS for minimal model (τ 0 (x, t) limit)
For the bulk 4-point function calculation, the computation proceeds in an identical manner, with the following operator insertion points on the CWS: ω 1 = ω 3 = iτ, ω 2 = ω 4 = x. As 23 As we have shown earlier, the TOC cannot encircle the η = 1 branch point and the contribution gives 1.

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we discussed in 4, in τ 0 (x, t) limit the bulk four-point OTOC on the UHP is factorized holomorphically and each factorized correlator is identical to OTOC in a thermal state with the identification β = 4τ 0 . Hence at CWS, we can safely use the result of thermal OTOC with that identification. As described in [9], the CFT 2 cross ratios η crosses the branch cut in the conformal block at Where ij ≡ i e 2π β i i − e 2π β i j . For large c CFT 2 , the second sheet effect of the vacuum Virasoro block gives the non-trivial OTOC as in (4.7) [9]. However in the unitary minimal model, the similar analysis leads to a universal constant value which is completely determined by modular S matrix [35,36]. However the analysis neglects the time dependence of approaching the constant value at late time. In [37], the author studied two different time regime: the early time regime t − x τ 0 and the late time regime t − x τ 0 . However a different correlator of the form 24 is used to diagnose such time dependence in different time regime. However this type of correlators provides a prescription for regulators which could violate our desired condition that bulk operators are placed very far away from the boundary. 25 Here we will see that we could still manage to find the late time dependence in approaching the constant value of minimal model OTOC using the standard prescription of iτ i → t + i i as we used before [9].
We would like to find the normalized OTOC of the following form for minimal model: As usual, we first consider the Euclidean correlator (A.4). In terms of holomorphic conformal blocksF p , F can be expanded as follows: Here p denotes the primary of dimension h p appeared in W W and V V OPE. In minimal model, there are finite number of such primaries. These blocks F p (η) have a nice series expansion in η(similarly for F p (η)).
Here K denotes the level of descendant and {K} is the collection of descendants of the conformal family given by a particular primary p. This F p (η) has a branch cut [1, ∞).
Since η crosses the cut at t = x, F p (η) picks a monodromy matrix F p (η) → p M pp F p (η).
If W W and V V OPE has multiple fusion channels, then M pp is a nontrivial matrix instead 24 This is closely related to MSS type correlator with F (t + i β 4 ) [10]. 25 In Euclidean, x τ0 τ is required to ignore boundary effect. However that particular choice of regulated OTOC could violate this condition which one can see easily.

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of a phase. This non-trivial monodromy dictates how much OTOC C 1 (t) differs from TOC C 2 (t) at late time.
In the late time regime t ± x τ 0 , (η,η) 1 from (B.1). If we choose the fusion channel as h p 1 < h p 2 < · · · < h pn and take the smallest primary h p 1 = 0 i.e. p 1 is the identity channel, we can expand C 1 (t) and C 2 (t) as the following way in η,η 0 limit: Similarly for TOC we get