An effective field theory for non-maximal quantum chaos

In non-maximally quantum chaotic systems, the exponential behavior of out-of-time-ordered correlators (OTOCs) results from summing over exchanges of an infinite tower of higher “spin” operators. We construct an effective field theory (EFT) to capture these exchanges in (0 + 1) dimensions. The EFT generalizes the one for maximally chaotic systems, and reduces to it in the limit of maximal chaos. The theory predicts the general structure of OTOCs both at leading order in the 1/N expansion (N is the number of degrees of freedom), and after resuming over an infinite number of higher order 1/N corrections. These general results agree with those previously explicitly obtained in specific models. We also show that the general structure of the EFT can be extracted from the large q SYK model.


Introduction
Information injected into a small subsystem of a quantum many-body system eventually spreads under time evolution across the entire system. Such scrambling of quantum information can be described in terms of growth of operators under Heisenberg evolution. More explicitly, consider a quantum mechanical system with N degrees of freedom and few-body interactions among them. The growth of operators can be probed by the so-called out-of-time-ordered-correlators (OTOC) [1][2][3][4][5][6] F (t) = ⟨W (t)V (0)W (t)V (0)⟩ β = ⟨Ψ 2 (t)|Ψ 1 (t)⟩, (1.1) Here V and W are generic few-body operators which we will take to be Hermitian, and ⟨· · ·⟩ β denotes the thermal average at an inverse temperature β. In (1.2), |Ψ β ⟩ denotes the thermal field double state the expectation values with respect to which give the thermal averages.
In the large N limit, the degrees of freedom involved in generic few-body operators V (0) and W (0) do not overlap with each other. For small t, V (0) and W (t) almost commute, and |Ψ 1,2 ⟩ are almost identical, which means that F (t) should be O(1). As time increases, W (t) grows, and Ψ 1,2 become more and more different, which decreases F (t). It is expected for chaotic systems [5] F (t) ∼ c 1 − c 2 N e λt + · · · (1. 3) where c 1,2 are some constants and λ is the quantum Lyapunov exponent. In contrast,
(1. 6) The bound is saturated by various systems, including holographic systems in the classical gravity limit, and SYK-type systems in the low temperature limit. These "maximally" chaotic systems are special: the exponential time dependence in (1.3) can be attributed to the exchange of the stress tensor between W and V (see Fig. 1a), and can be described by a hydrodynamic effective theory with a single effective field φ that plays the dual role of ensuring energy conservation and characterizing operator growth [11,12]. For non-maximal chaotic systems with λ < 2π β , the origin of (1.3) is more intricate, arising from exchanging an infinite number of operators. For example, in the SYK system, exchange of an operator characterized by some quantum number j leads to an exponential decrease in F (t) proportional to e 2π β (j−1)t , which violates the bound for any j > 2. Summing over exchanges of an infinite tower of such operators with increasingly larger values of j leads to an effective λ satisfying the bound. We will loosely refer to j as "spin" in analogue with higher dimensional systems even though there is no spin for SYK. Another example is four-point correlation functions of a large N CFT in the vacuum state in the so-called conformal Regge regime [13][14][15], which can be interpreted as a thermal OTOC in terms of Rindler time (with β = 2π). Here contribution from a spin-j operator in the OPE of W W (and V V ) gives a contribution proportional to e (j−1)t (t is now the Rindler time) and summing over an infinite number of higher spin operator exchanges gives an λ < 1. It is natural to ask whether there exists an effective description with a small number of degrees of freedom that can capture the sum over exchanges of the infinite tower of operators, with j eff = 1 + λ β 2π interpreted as the effective spin of the effective fields. We should stress that such an effective description is conceptually and philosophically different from that is usually used in effective field theory (EFT). Usually an EFT is used to describe the dynamics of a small number of "low energy" (or "slow") degrees of freedom whose contributions dominate over others in the regime of interests. Their effective actions can be formally defined from path integrals by integrating out other "high energy" (or "fast") modes. There is, however, no such decoupling of "high energy" (or "fast") degrees of freedom here. Spin j (with j > 2) exchanges give important contributions to F (t); they cannot be integrated out in the usual sense. We are merely asking whether there is a way to capture the effect of the infinite sum. The effective fields here may not correspond to genuine physical collective degrees of freedom. The philosophy is also very different from the EFT for maximal chaos described in [11]; there the stress tensor exchange dominates and the EFT is used to capture the most essential part of the stress tensor exchange.
The question of an effective description for non-maximal chaotic system is closely related a well-known problem in QCD, the formulation of EFTs for Reggeons (see e.g. [16] for a review). Consider a scattering process in some quantum system (say in QCD or string theory) where V, W denote different particles. We denote the scattering amplitude by A(s, t), with s the standard variable characterizing the total center of mass energy, and t characterizing the momentum exchange between V and W particles. In the regime s → ∞ with t finite, each spin j particle exchange between V and W gives a contribution to A(s, t) proportional to s j−1 . When the Sommerfeld-Watson transform is used to summing over all the higher spin exchanges, the scattering amplitude can be written in a form A(s, t) ∝ g W W (t)g V V (t)s α(t)−1 (1.8) which can be interpreted as the exchange of a single effective particle, called reggeon, with an effective spin α(t). g W W , g V V can be interpreted as couplings the Reggeon to W and V . See Fig. 1b for an illustration. For systems with a gravity dual, the OTOC (1.1) maps to the gravity side a scattering process precisely of the form (1.7) in a black hole geometry [1][2][3][4], with V, W the corresponding bulk particles dual the boundary operators. The center of mass energy square s for the scattering process is related to time separation t in (1.1) by s ∝ e 2π β t . In the bulk language, λ arises from summing over exchanges of an infinite number stringy modes with increasingly higher spins.
In the α ′ → 0 limit, the contributions of higher spin stringy modes decouple, with only graviton exchange remaining, and the system becomes maximally chaotic. In this limit, the maximal value λ max = 2π β is universal for all holographic systems, independent of the details of the black hole geometries [10], and can be argued as a direct consequence of existence of a sharp horizon.
Having an effective description away from the α ′ → 0 limit that can capture an infinite number of stringy modes exchanges is clearly valuable. Such an effective description can also potentially give insights into what becomes of the event horizon in the stringy regime.
In this paper we make a proposal to formulate an EFT for a non-maximal chaotic system. For simplicity, we will restrict to a quantum mechanical system with no spatial dependence. Generalization to having spatial dependence should be straightforward, and will be left elsewhere. Lacking at the moment a first-principle understanding of the nature of the effective chaos field(s) or their effective action, our approach is phenomenological. We try to identify a minimal set of fields and a minimal set of conditions on their action, such that the following criteria are met: 1. With λ being an input parameter, the EFT gives rise to exponential behavior (1.3) for OTOCs, but no exponential for TOCs (1.4)-(1.5).

It captures all the KMS properties of thermal 4-point functions.
We will see that the above conditions are rather constraining, and the resulting EFT can be used to make a general prediction on the structure of OTOCs, which is consistent with that previously postulated in [17,18], and agree with the explicit expressions in large q SYK model, holographic systems (obtained from stringy scattering), and the conformal Regge theory. Furthermore, we show that in this framework it is possible Figure 2. Effective description of W (t 1 )W (t 2 ) in non-maximal chaotic systems. The black dots are bare operators W 0 . There are now two types of dressing: one type dresses each local operator separately (yellow "clouds"), and the other type dresses both operators together (blue "clouds"). Maximal chaos case contains only the first type of dressing.
to sum higher order terms in equation (1.3) of the form e kλt N k (with k an integer) 2 in an exponential, which can again be viewed as a general prediction, and agrees with those previously obtained in specific systems [4,19,20].
We also show that the structure proposed for the non-maximal chaos EFT can be in fact extracted in the large-q SYK model, where it is possible to identify explicitly the chaos effective fields, and make much finer comparison between the EFT and the microscopic theory than the structure of TOCs and OTOCs.
It is worth mentioning here a key difference between the non-maximal EFT to be discussed in this paper and that for maximal chaotic systems of [11,12]. For maximal chaotic systems, W (t) can be viewed as W (t) = W [W 0 (t), φ(t)] where "bare" operator W 0 (t) describes W (t) in the large N limit (with no overlap with V 0 ). φ(t) is an effective field "attached" to W 0 , i.e. W is obtained by dressing W 0 with φ. φ captures effectively the overlap between W (t) and V (0) due to scrambling, and its dynamics leads to 1/N corrections indicated in (1.3). For non-maximal chaotic systems, the chaos fields involve multiple components: (i) one component dresses each bare local operator as in the maximal chaotic case (and indeed it reduces to φ in the maximal chaos limit). This component carries only one time argument. (ii) There are other components which dress both W 's in (1.1), i.e. it has two time arguments (see Fig. 2 for an illustration). Existence of such components leads to many new elements which are not present in the maximal case.
The paper is organized as follows. In Section 2, we construct the effective field theory of non-maximal chaos with two effective fields ϕ 1,2 and show that the TOC does not have exponential growth and OTOC has exponential growth. In Section 3 we compare the general structure of OTOCs obtained in Sec. 2 with various known examples. In Section 4, we show that the two effective fields ϕ 1,2 reduce to a single field in the maximal chaos limit and the EFT becomes the same as the EFT constructed for maximal chaos. In Section 5, we show how the general structure of the EFT discussed in Sec. 2 arises in the large q SYK model, and obtain the explicit form of the EFT action. In Section 6, we include higher order coupling to effective mode ϕ 1,2 and show that certain higher-order terms of the four-point function can be resummed and exponentiated. We conclude in Section 7 with a summary and a discussion of future directions.

The structure of EFT
In this section we discuss the general formulation of an EFT for non-maximally chaotic systems. For simplicity we will consider quantum mechanical systems with no spatial dependence. Generalization to systems with spatial dependence can be readily made, although technically more intricate. We will also use the unit such that β = 2π .

General setup
Consider a generic four-point Wightman function in thermal state where the subscript refers to the ordering of operators and the time argument should be understood as corresponding to each subscript in the same order. We will treat time variables as complex, and F abcd (t 1 , t 2 , t 3 , t 4 ) is analytic in the domain which can be iterated cyclically to shift other time variables. It is convenient to introduce a time-ordered function where T denotes operators should be ordered from left to right according to the ascending order of their corresponding ℑt i . Moreover, it should always be understood that The KMS condition (2.3) can then be written aŝ where m i are integers, and should be such that t ′ i 's obey (2.5). Now considerF which by definition is symmetric under swapping t 1 ↔ t 2 and t 3 ↔ t 4 can correspond to TOC or OTOC. For example, Our goal is to develop an effective description for obtainingF W W V V for large N . To motivate the structure of our proposed EFT for non-maximally chaotic systems, it is useful to recall some key elements of that for maximally chaotic systems introduced in [11]. One imagines the scrambling of W (t) allows a coarse-grained description in terms of building up an "effective cloud," i.e. (2.11) Here W 0 (t) is a "bare" operator involving the original degrees of freedom of W , and φ(t) is an effective chaos mode that describes macroscopically the growth of the operator in the space of degrees of freedom. W (t) in (2.11) is taken to be linear in W 0 but can in principle have any dependence on the effective field φ(t). The dynamics of φ is governed by a chaos effective theory, with two-point function of φ scaling with N as 1/N . Thus W 0 can also be viewed as giving the leading part of W (t) in a 1/N expansion. When φ(t) is small, it can be expanded to linear order as where L t [W 0 ] is a W 0 -dependent differential operator acting on φ. More explicitly, Below it should always be understood that L t acts on the corresponding φ(t) even when they are not written adjacently. It then follows that where we have assumed that one-point function of φ is zero. The O(1/N ) piece in the above equation comes from O(φ 2 ) term in (2.14), and is proportional to two-point function of ϕ. The KMS condition for g W is as for generic few-body operators V, W , their two-point function should vanish at the leading order in 1/N expansion. Plugging (2.14) and the corresponding expression for V into (2.7),F W W V V reduces to the two-point function of effective mode φ(t) at leading order whereL t is the differential operator from a similar expansion of V with c mn →c mn , and Here the ⟨·⟩ EFT means expectation value evaluated in the effective field theory of φ; T in ⟨T φ(t i )φ(t j )⟩ EFT follows from the relative magnitude of ℑt i and ℑt j . Equation (2.17) has a very restrictive structure: the two-point function of φ in each term only depends on the locations of two operators. For example, the φ correlation function ⟨T φ(t 1 )φ(t 3 )⟩ EFT has no knowledge of t 2 , t 4 at all. In other words, the four-point function essentially reduces to pairwise two-point functions of φ. This structure leads to various features ofF that are consistent with a maximally chaotic system [12], including the Lyapunov exponent λ = 1 (after imposing a shift symmetry in the EFT of φ), but are not present in a non-maximally chaotic system. Since (2.17) is a direct consequence of (2.12), for non-maximally chaotic systems, we must generalize (2.12).

Two-component effective mode and constraints from KMS symmetries
We will now propose a formulation for non-maximally chaotic systems which may be considered a minimal generalization of the EFT in [11] for maximal chaos. The formulation is partially motivated from features of the large q SYK theory, and as we will show in Sec. 5, fully captures the physics of that theory. The general structure of OTOCs resulting from it is also compatible with the conclusions of [4,[13][14][15]21], as we will describe later.
In this formulationF still reduces to two-point functions of some effective fields, but the main new ingredient we would like to incorporate is that now two-point functions of effective field(s) have knowledge of the locations of all four operators, not just two of them. For this purpose we consider the following generalization of (2.12) 3 (2.19) where there are two fields ϕ 1,2 who depend on both t, t ′ , and D (1,2) W (t, t ′ ) are some W 0 -dependent differential operators to be specified more explicitly below. Now ϕ 1,2 depend on both t and t ′ of W (t) and W (t ′ ), which means that we cannot view ϕ 1,2 as the "dressing" of each individual operator, as in the case of (2.12). Rather they should be interpreted as an effective description of the sum over an infinite number of higher spin operator exchanges that are known to contribute toF at the leading order in non-maximal systems [4,[13][14][15]. There is a parallel equation with W replaced by V 's.
The effective theory of ϕ i should satisfy the following criteria: 1. Exponential growth in OTOCs with an arbitrary Lyapunov exponent λ.
2. No such exponential growth in TOCs.
3. All the KMS conditions and analytic properties ofF W W V V are satisfied.
We will show that the above goals can be achieved with a minimal generalization of (2.14). In this subsection we first present the prescription for (2.19), and work out the constraints on two-point functions of ϕ 1,2 from the KMS conditions ofF W W V V , which provide the basic inputs for formulating the theory of ϕ 1,2 . We will take ϕ 1,2 to "mainly" couple to one of the W 's. A definition which respects the swap symmetry (2.8) ofF is that ϕ 1 (ϕ 2 ) couples mainly to the W with the smaller (larger) ℑt. Denoting t S (t L ) with the smaller (larger) value of ℑt, ℑt ′ , by "mainly" we mean: 1. ϕ 1 (t, t ′ ) = ϕ 1 (t; t S ) depends weakly ont = t+t ′ 2 such that it can be expanded in terms oft-derivatives. The dependence ont encodes the nonlocal information of the theory. Similarly, ϕ 2 (t, t ′ ) = ϕ 2 (t; t L ).

The action of D
In other words, ϕ 1 couples directly only to W (t S ), but does feel the presence of W (t L ) through weak dependence ont. Note that equation (2.20) should be understood to be valid within time-ordered correlation functions, thus there is no need to worry about orderings between W 0 (t L ) and W 0 (t S ). Equation (2.20) contains no derivative with respect tot; it can be viewed as the leading term in a derivative expansion oft.
For notational simplicity we take the coefficients c nm in (2.20) to be the same for D (2) W (t, t ′ ), but our discussion can be straightforwardly generalized to the cases that they are not the same. The vertex for V will be denoted asL t with c mn →c mn . The above prescription is a minimal nontrivial generalization of (2.12) that satisfies the aforestated criteria. In Appendix A we show that a few other simpler prescriptions cannot work.
Since in (2.19) |ℑt − ℑt ′ | < 2π, by definition, ϕ 1 (t; t) is defined for ℑt − ℑt ∈ (0, π), while ϕ 2 (t; t) is defined for ℑt − ℑt ∈ (−π, 0). We refer to their domains as I 1 and I 2 respectively, see Fig. 3. Due to symmetries in exchanging t 1 and t 2 , we will take ℑt 1 < ℑt 2 , and similarly take ℑt 3 < ℑt 4 . Therefore, ϕ 1 always mainly couples to W (t 1 ) and V (t 3 ), ϕ 2 always mainly couples to W (t 2 ) and V (t 4 ). Substituting (2.19 wheret W = (t 1 + t 2 )/2,t V = (t 3 + t 4 )/2, and the expectation value of ϕ 1,2 should be understood as being evaluated in an effective theory. Unlike in (2.17), where the timeordering T follows from that ofF, here in (2.23) the effective fields ϕ 1,2 (t; t) have two time variables, and time-ordering inF no longer leads to a unique choice of orderings of ϕ i and ϕ j . We will specify the precise meaning of T ϕ i (t W ; t i )ϕ j (t V ; t j+2 ) in Section 2.3. Here we will just list some properties they should satisfy: 1. Since in time ordered correlation functionF we can exchange V and W arbitrarily, the ordering of ϕ i , ϕ j in the correlation function should not matter, i.e.
(2.24) Figure 3. The green region is the domain I 1 for ϕ 1 (t; t) and the blue region is the domain I 2 for ϕ 2 (t; t). The KMS conditions (2.26)-(2.27) relate ϕ 1 with ϕ 2 through identifying points between I 1 and I 2 as indicated by the black arrows. This generates the periodicity (2.29) on I 1,2 (red arrows).

2.
From time translation invariance of the system,F is invariant under shifts of all t i by the same constant, which implies the following translation invariance of two-point functions of ϕ 1,2 , 3. The KMS conditions satisfied byF imply that these two-point functions of ϕ 1,2 should satisfy the following constraints where ≃ means equal up to zero modes, defined as functions n ij (t 1 , t 2 ; t 3 , t 4 ) satisfying The zero modes can be viewed as field redefinition freedom of effective fields that does not cause any difference in the original four-point functionF. From now on, we will set n ij = 0. Combining (2.26)-(2.27), we also get the following periodicity 4. Four-point functionF W W V V (t 1 , t 2 ; t 3 , t 4 ) can have potential non-smoothness when the imaginary parts of two or more time arguments coincide, as these are the locations where ordering of operators change. There are two cases: (a) ℑt 1 = ℑt 2 , which corresponds to order changes of W 's within themselves. In terms of ϕ 1,2 (t; t), this corresponds to ℑt−ℑt = 0, where the couplings of W 's to ϕ 1,2 are switched. 4 Similar statements apply to t 3 , t 4 . As stated earlier, we will restrict to ℑt 1 < ℑt 2 and ℑt 3 < ℑt 4 throughout, so such potential non-smoothness will not be relevant for our discussion of T ϕ i (t; t)ϕ j (t ′ ; t ′ ) .
(b) One of ℑt 1 , ℑt 2 coinciding with one of ℑt 3 , ℑt 4 , which is a boundary between the domains of t i corresponding to TOCs and OTOCs; crossing such a boundary a pair of W and V will exchange order. In terms of two-point function T ϕ i (t; t)ϕ j (t ′ ; t ′ ) , this corresponds to potential non-smoothness at ℑt = ℑt ′ . In the domain D, we should not have any other singularities.

Diagonalize the KMS conditions
We will now proceed to formulate an effective field theory (EFT) that can be used to obtain correlation functions of ϕ i in (2.23).
There is an immediate difficulty in directly formulating an EFT for ϕ 1,2 , due to that they are defined in different domains (recall Fig. 3). So they cannot appear in the same Lagrangian, but they transform to each other under the constraints (2.26)-(2.27) from the KMS conditions. To address this difficulty, we introduce two new fields, which are both defined in the domain I 2 : ℑ(t − t) ∈ (−π, 0). Conversely, we have We will define two-point functions of ϕ 1,2 in (2.23) in terms of those of η ± using (2.31). For example, where on the right hand side the time orderingT is defined in terms of that of ℑt, i.e.
Now ⟨· · ·⟩ is understood as defined in the EFT of η ± , and the right hand side of (2.33) should be understood as Wightman functions in the EFT. The motivations for choosinĝ T ordering in terms of ℑt are as follows. Firstly, as discussed in item 4b at the end of last subsection, correlation functions of ϕ 1,2 have potential non-smoothness at ℑt = ℑt ′ . Ordering in ℑt in η-correlators provides a simple way to realize that. Secondly, we assumed that the dependence of ϕ 1,2 ont is weak, so should be η ± . Making the ordering independent oft is natural. Now consider the constraints (2.26)-(2.27). It can be checked that they are satisfied provided that which is diagonal in η ± . We see that introducing η ± not only resolves the domain issue, but also diagonalize the constraints from KMS conditions. Equation (2.34) implies It then follows from (2.35) that i.e. time-ordered functions of η ± are also diagonal. We now proceed to formulate an effective theory of η ± with the following considerations in mind: 1. With the assumption of weak dependence ont, we assume that the effective action can be expanded in derivatives oft. This leads to an immediate simplification: with only derivative dependence ont, the EFT becomes translationally invariant int. Now given (2.36), we also have translation invariance in t, i.e. The domain for function is then given by the shaded stripe indicated in Fig. 4.
At quadratic order in η ± , the effective action should be translationally invariant in botht and t.
2. We would like to interpret (2.34) as the KMS conditions for the η ± -system at a finite temperature. Given the definition ofT in terms of t, it is natural to interpret the temperature as being associated with t. However, the condition (2.34) shifts botht and t simultaneously, which is not of the conventional form. In next subsection, we will discuss how to convert it into the standard form.
3. So far the time variablest, t are complex. To write down an effective action we need to choose a real section in the complext, t planes. From Fig. 4 it is convenient to choose the section to be that of imaginaryt and real t, i.e. we will lett = −iτ and write down an action for η ± (τ ; t). It can be viewed as a twodimensional field theory withτ being a "spatial" coordinate and (real) t being time. Behavior of correlation functions for η ± elsewhere are obtained by analytic continuations.

Reformulating the KMS conditions
In this subsection we reformulate (2.34) as the KMS conditions for η ± at a finite temperature (associated with t) withτ = −ℑt as a spatial direction. Let us first recall the standard story. For a quantum field χ in a two-dimensional spacetime (t,τ ) at a nonzero inverse temperature β, the KMS condition for Wightman functions are ⟨χ(τ 1 , t 1 )χ(τ 2 , t 2 )⟩ = ⟨χ(τ 2 , t 2 )χ(τ 1 , t 1 + iβ)⟩, and the Feyman functions G F (τ , t) ≡ T χ(τ , t)χ(0, 0) = G F (τ , t + iβ), with its fundamental domain being ℑt ∈ (−β, 0). G F (τ , t) may have non-analytic behavior such as branch cuts at ℑt = 0 and ℑt = −β. Now consider G F (t; t) defined in (2.39). From (2.34), the fundamental domain of G F (t; t) can be chosen to be the region D η in Fig. 4. Equation (2.34) is not quite the KMS condition with β = π (note that this is 1 2 of the temperature we started with) due to the shift inτ . We can resolve this issue by extending the region D η to the larger region D * in Fig. 4. Region D η is bounded above and below by lines τ − ℑt = ±π, which are part of the boundary of the analytic domain of the original four-point functionF. The behavior of G F at these boundaries are system-dependent and depend on UV physics. In other words, in principle for different systems different boundary conditions should be imposed there. In the spirit of effective field theories we expect that the general structure of the effective action should not depend on the specific UV physics, although the coefficients in the effective action will. Since we are only interested in the general structure of the effective action, we can choose a most convenient boundary condition: we extend the domain to D * , and identify the values of G F (t; t) atτ = −π andτ = 2π. In other words, we have periodic boundary conditions inτ direction. Note that later we will only need to use the behavior of G F (τ ; t) in region D η .

The quadratic effective action
In this section, we will construct an effective action for η s,p (τ ; t) defined in last subsection. We treat Euclidean timeτ as spatial coordinate in the rangeτ ∈ [0, π] and t as real time. η s,p (τ ; t) satisfy the boundary conditions (2.41) inτ direction. Real-time action for excitations in a thermal state can be formulated using the Schwinger-Keldysh formalism. We will follow the non-equilibrium EFT approach developed in [11,[22][23][24].
To write down a real-time action we need to double the degrees of freedom on a two-way Keldysh contour for t, where the fields η (1) s,p and η (2) s,p are on the first and second contour respectively (see Fig. 5). For each η s,p we also have the so-called r-a variables η r s,p , η a s,p defined as The effective action should satisfy various unitary constraints and the dynamical KMS condition (to ensure local thermal equilibrium). We derive these conditions in detail in Appendix B, and just briefly present them here.
1. The action S[η r s,p , η a s,p ] should contain terms in the form of 2. Each term in above form must contain at least one η a s,p .
4. For the terms with odd numbers of η a s,p , the action needs to be real, which means for α 1 , · · · , α k contain odd numbers of a. At quadratic order, the effective action S EFT can then be written as

The action needs to obey dynamical KMS condition
where from the above conditions we have In Appendix B, we derive the following dynamical KMS condition for the quadratic action (2.60) As shown in [25], setting K aa s,p = 0 means the local entropy current is conserved and the system is non-dissipative. In this case, (2.63) reduces to and the resulting action can be factorized [12,24,25] Taking t → −iτ in the above action we obtain a Euclidean action defined for both Euclidean timesτ , τ . We stress that the factorization and thus the Euclidean action are not possible when dissipations are included. For simplicity, in this paper we will only consider the non-dissipative case with constraint (2.64) though the generalization to dissipative case should be straightforward.
As discussed earlier, we assume that the action can be expanded in derivatives of τ . As in [11,12], we cannot, however, expand the action in derivatives in t, since we are interested in time scales of order 1/λ so as to be able to probe the exponential growth e λt . The Lyapunov exponent λ could be comparable to the inverse temperature β, and thus there is no scale separation in t.
Since η s,p (τ ; t) have different boundary conditions (2.41), they allow different lowest order of ∂τ in the action. For η s,0 , which is periodic inτ , the lowest order of ∂τ in K ar For η s,± , which gains a nontrivial phase after shiftτ →τ + π, the lowest order of ∂τ in K ar s,± must be nontrivial, and we will keep to the linear order It follows from (2.64) that for all p, Thus K s,0 (x) is an even function of x with real coefficients (when expanded in power series), while K s,± (x) are functions of pure imaginary coefficients.
Keeping only leading orders, we can reduce K ar s,0 piece to one dimension of t and write the leading order quadratic effective action as (2.69) With the leading order effective action (2.69), we have Give the periodic boundary condition (2.41), we can write where ∆ s,p (t) can be written in Fourier space as which holds for t > 0. Here the integral contour C must be above all poles of integrand on the complex ω plane because G ra s,p (τ ; t) is proportional to θ(t). Note that θ(τ ) in (2.73) comes from ∂τ in (2.71).
Equations (2.73) imply that except for certain jumps atτ = 0, correlation functions have no dependence onτ . From item 4b, however, such branch cut should not be present in the four-point functionF, and thus should be cancelled in (2.23), i.e.
for infinitesimal positive ϵ. Also note that the above condition is relevant only for TOC of types ⟨W V V W ⟩ and ⟨V W W V ⟩ for which ℑt W −t V could have either sign without changing the order of four fields. All other four-point functions have definite sign for ℑt W −t V . Now recall the smoothness conditions (2.56), which upon using (2.73)-(2.75) leads which shows that the terms in (2.69) are not independent. It can also be checked that (2.77) is consistent with constraints (2.68).

Shift symmetry and exponential growth in correlation functions
Similar to the maximal chaos case [11], we will postulate that the action and the vertices ( where α ± are constants.
2. There is no exponential growth in the symmetric correlation functions of η + , i.e.
Note that the KMS condition (2.48) leads to the fluctuation-dissipation relation In terms of ϕ 1,2 , the shift symmetry (2.78) can be written as We also require that the vertices in (2.20)-(2.21) be compatible with the shift symmetry (2.78), i.e. L t satisfies (2.82) Figure 6. The contour C on the ω plane for retarded propagator G ra s,p (τ ; t). The two red crosses are poles at ω = ±iλ due to shift symmetry.
Since η − is a sum of η −,0 , η −,± , the invariance under (2.78) means that at least one of K −,p has a factor of ∂ 2 t − λ 2 . For convenience, we will write The constraint (2.77) implies that K +,p may also contain a factor ∂ 2 t − λ 2 , and we can similarly The general structure of our discussion will not depend on the specific forms of k +,p and k −,p . Following from (2.74), we have where the contour C is chosen to be above all poles (see Fig. 6) of the integrand because the LHS is proportional to θ(t). We then find 8 where we used the parity of k s,0 , and · · · denote possible contributions from other singularities. From now on we will suppress · · · and only write exponential terms. Using (2.80) and Similarly, from (2.75) we have where h s,± are given by The constraint (2.79) implies that, up to non-exponential pieces, which upon using (2.88) and (2.92), further implies Note that (2.94) is consistent with (2.68). Notice that the factor tan π 2 (λ − 1/3) on the right hand side of (2.94) for k +,− (iλ) becomes zero for λ = 1/3, which cannot happen as a zero for k +,− (iλ) would lead to divergences in (2.92). This means that k +,0 must have a pole at λ = 1 3 , i.e. k +,0 (iλ) ∼ (λ − 1/3) −1 , which in turn means that the prefactor in (2.86) vanishes and that the factor ∂ 2 t − λ 2 in K +,0 is in fact not there (it cancels with a factor hidden in k +,0 ). For λ = 2/3, the right hand side of (2.94) is divergent for k +,+ (iλ), which means that the factor ∂ 2 t − λ 2 should also cancel for K +,+ (i∂ t ) at λ = 2/3. The divergence of the factor tan πλ 2 for λ = 1 will be commented on later in Sec. 4.

Summary of the effective field theory
We have now discussed all elements of the EFT formulation, which we summarize here in one place: 1. The product W (t 1 )W (t 2 ) is written in terms of an expansion in terms of two effective fields ϕ 1 (t W ; t S ) and ϕ 2 (t W ; t L ) through a vertex. Similar expansion applies to V (t 3 )V (t 4 ). At leading nontrivial order in the 1/N expansion, we havê (2.95) The domain of ϕ i is given by I i in Fig. 3.

The KMS conditions ofF impose constraints on correlation functions of ϕ i , which
can in turn be obtained in terms of those a new pair of fields 3. The effective action of η ± is written for pure imaginaryt = −iτ and real t. Correlation functions of η ± for general complext and t are obtained from analytic continuation. We also assume that the effective action can be expanded in terms of derivatives ofτ , which in turn implies that the action is translation invariant for bothτ and t. Two-point functions of η ± are then defined in the domain D η of Fig. 4.
The domain D η is irregular and inconvenient to work with. It is extended to D * of Fig. 4. η ± is then further decomposed into in terms of periodicity conditions inτ -direction With the decomposition (2.97)-(2.98), the KMS conditions of the original fourpoint functionF can be formulated in terms of KMS conditions for η s,p at the inverse temperature π (half of the original inverse temperature), and can be written as periodic conditions in the imaginary t direction The leading actions in theτ -derivative expansion for η s,0 contain noτ derivative and thus η s,0 can be thought asτ -independent, while the leading actions for η s,± contains oneτ -derivative. 5. From (2.96), (2.97) and (2.99), we can write ϕ 1,2 as where we have used that η ±,0 can be viewed as being independent ofτ . Note from (2.99) (2.103) 6. For OTOCs to have exponential dependence on t, we impose the shift symmetry on both the action and the vertex. We also require no-exponential growth in G rr + (τ ; t), which is needed such that TOCs do not have exponential t-dependence. This condition requires that various terms in the action should obey (2.94).

TOC and OTOC
Now consider the following two four-point functions where ℜt 1 , ℜt 2 ≫ ℜt 3 , ℜt 4 or ℜt 1 , ℜt 2 ≪ ℜt 3 , ℜt 4 , i.e. F 4 is OTOC and G 4 is TOC. We suppose each t i has a small imaginary part such that the orderings in (2.105) follow that defined in (2.4) (see Fig. 7 as an illustration). For F 4 , the small imaginary part for each t i leads to ℑt W < ℑt V , but for G 4 , depending on the relative value of the imaginary part of each t i , we may have either ℑt W < ℑt V or ℑt W > ℑt V . For definiteness, we consider the former case ℑt W < ℑt V .From (2.23) and (2.49)-(2.51), we find that where we have used (2.54), and In the second line of the above equation we have used (2.94). We thus find that the difference between OTOC and TOC has exponential growth. Note that there is no divergence in (2.107) at λ = 1 3 ; as mentioned earlier below (2.94), k +,0 (iλ) has a pole at λ = 1 3 , which is canceled by 1/2 − cos πλ. We will now show that TOC G 4 does not have exponential growth. In terms of Wightman functions (2.49) to (2.51), G 4 can be written as where we assume ℑt W < ℑt V and ± sign in the first time argument means ℑt > 0 or ℑt < 0. Using (2.88), (2.91), and (2.82) (and the counterpart forL t ), we can simplify (2.108) as It can be checked using (2.94) that, C 1 = C 2 = 0. We can similarly examine G 4 for ℑt W > ℑt V and another type of TOC with ℑt W < ℑt V . We again find their exponential growth pieces vanish due to (2.94). In particular, the condition (2.76) is automatically satisfied up to non-exponential pieces. Given that TOCs do not have exponential terms, equation (2.107) implies that the exponential terms in OTOCs depend only on k +,0 (iλ). Note that k +,± (iλ) are determined from k +,0 (iλ) by (2.94), and k −,p are also constrained from k +,p from (2.77).
In (2.85) and (2.89), we assumed for simplicity that the integrand only has simple poles at ±iλ. This assumption can be relaxed to have higher order poles. In fact, it can be shown that at most double poles are allowed due to the condition (2.76). These double poles lead to linear-exponential terms te ±λt in the correlation functions of η ± . Interestingly, the contributions from the double poles to any four-point function cancel out. So what we discussed in fact gives the most general form for four-point functions. See Appendix C for details.

General structure of OTOCs for non-maximal chaos
We have seen that the shift symmetry (2.81) and requirement (2.79) guarantee exponential growth of OTOC and the absence of exponential growth of TOC. We will now examine the general structure of OTOCs as predicted by the theory.
Using (2.13), we can expand (2.82) explicitly as Similar to [12], we define and (2.114) becomes where we used invariance of g W under KMS transformation. This is compatible with (2.117) without any restriction on λ, unlike the EFT of [11], where the KMS condition of g W restricts λ = λ max = 1 [12]. Using the definition in (2.115) and (2.116), we can write the OTOC F 4 in a more symmetric way. Since TOC G 4 does not have exponential piece, OTOC F 4 has the same exponential piece as (2.106). Using the shift symmetry of vertex (2.82), we can write each exponential in a symmmetric way Using (2.115) and (2.116), we can write the connected piece where we have used α to denote the prefactor of (2.107), i.e. (2.107) becomes and used (2.117) in the last line. Equation (2.122) has the same structure as that assumed in [17,18], including the phase e iπλ/2 . In [17,18], in the place of G W even (λ, t) and G V even (−λ, t) are certain advanced and retarded vertices, which are invariant under KMS transformation t → −t−2πi. For a specific microscopic system, these two vertices may obey certain differential equations [17,18], which in our language translate into conditions on the effective vertices. In (2.122), there is a second term, obtained from λ → −λ, which exponentially decays for t 1 , t 2 ≫ t 3 , t 4 , and was not present in [17,18].
Here it is a consequence of shift symmetry for both signs in (2.78), which are needed in order for TOCs to not have exponential growth. In the non-dissipative case, this term should also exist even if we only assume shift symmetry for just plus sign in (2.78) as explained in footnote 7. Now consider the double commutator defined as which can be rewritten in terms of four-point functions leads to (2.126) The factor cos λπ 2 → 0 as λ → 1, consistent with the result of [12] and those of SYK and holographic systems in the maximal chaos limit [4,17,18,26].

Comparisons with OTOCs in various theories
In this section we compare the general structure of OTOCs obtained in last section with various known examples.

The large q SYK model
We first look at the large q SYK model [27][28][29][30]. OTOC F 4 of fundamental fermions was obtained in [31] and has the form (after analytic continuation to Lorentzian signature) where G ψ (t) is the two-point function fundamental Majorana fermions in the large q SYK model (see more details in Section 5.1) given by where ∆ = 1/q is the conformal weight of the fundamental fermion ψ. The OTOC (3.1) has exactly the form of (2.122), with the cosh λ(t 1 + t 2 − t 3 − t 4 + iπ)/2 term containing exponentially growing and decaying terms that are both present in (2.122), including the phases. We can further identify As the simplest possibility we take c 1,n = δ n,0 and c 0,n = ∆δ n,1 which gives

Stringy scattering in a AdS black hole
The next example is the scattering in the AdS black hole background with stringy correction [4]. Assume the bulk spacetime dimension is d + 1 and the d-dimensional boundary coordinate is scattering near the horizon of the AdS black hole. At G N ∼ 1/N order, the OTOC F 4 is given by 9 where a 0 is a number depending on the background, s = a 0 p u p v is the total energy of the scattering in the unboosted frame, ψ 1,2 (p u , ⃗ z; x µ 1,2 ) are wavefunctions of W (x µ 1,2 ) expanded in the null momentum basis p u , ψ 3,4 (p v , ⃗ z; x µ 3,4 ) are wavefunctions of V (x µ 3,4 ) expanded in the orthogonal null momentum basis p v , ⃗ z and ⃗ z ′ are d−1 dimensional bulk transverse coordinates, and the integral runs over p u , p v , ⃗ z, ⃗ z ′ . With stringy correction, the scattering amplitude δ(s, |⃗ z|) for t 1 + t 2 ≪ t 3 + t 4 is given by where r 0 is the horizon radius, α ′ = ℓ 2 s (with ℓ s the string length) and
By translation symmetry of the AdS-Schwarzschild black hole in transverse directions, ψ i (p, ⃗ z; x µ ) are functions of ⃗ z − ⃗ x. By time translation symmetry, one can show that the wave function ψ i (p, ⃗ z; x µ ) in the unboosted frame has the following property which implies that ψ 1,2 = e 2πt/β f 1,2 (p u e 2πt/β ) and ψ 3, Here β is the inverse temperature of the black hole. Switching to the boosted frame by redefining p u → p u e −π(t 1 +t 2 )/β leads to which is a function of t 12 . Similarly we can redefine p v → p v e π(t 3 +t 4 )/β and find the wavefunctions for V become a function of t 34 . Taking this into (3.6), we can derive In the regime that the scattering amplitude δ(s, |⃗ z|) is of order one and slowly varies with respect to p u and p v , we can assume the integral over p u and p v in (3.11) can be approximated by their characteristic values p u c and p v c , which only depend on the wave functions. Moreover, the spatial dependence of wave function ψ i should be peaked around ⃗ z ∼ ⃗ x. To compare with our 0+1 dimensional EFT, we should integrate over all ⃗ x, which basically sets ⃗ x ∼ ⃗ z. Since wave functions are translational invariant along tranverse directions, we can integrate over ⃗ z directly in (3.7), which fixes ⃗ k = 0 and leads to where c d is a real constant and the Lyapunov exponent λ is It follows that we can write (3.6) as where f c W,V means f W,V taking value at p u,v = p u,v c and we have suppressed all transverse coordinates. Comparing (3.16) with (2.122), we see that they both have non-maximal exponential growth and the phase −iλβ/4 in (3.16) exactly matches with −iλπ/2 (of the −λ term) in (2.122) by β = 2π. In the case of t 1 + t 2 ≫ t 3 + t 4 , we need to flip the phase e −iπ/2 to e iπ/2 in (3.7) and will find consistency with (2.122) as well.
Matching (3.16) with (2.122) also leads to (3.17) For example, in AdS 3 [4], we have (β = 2π) where we have set all transverse coordinates as zero due to the integral over these directions. For large ∆ W , the characteristic p u is at p u c = 2∆ W −1 4i sinh(t 12 /2) , which leads to where g W (t) is the boundary two-point function in a thermal state. Note that the wave function (3.18) is computed in the pure gravity background and does not include any stringy corrections that should introduce non-maximal λ to the wave function. Thus (3.19) should be compared with the right hand side of (3.17) at leading order in α ′ expansion, i.e. we can identify G W even (1, t) = g W (t). It is clearly of interests to understand α ′ corrections of (3.18). In general dimension, the explicit forms of wave functions are not known, but (3.17) gives a constraint on their general structure.

Conformal Regge theory
The conformal Regge theory was developed in [13,14] and analyzed for Lyapunov exponent and butterfly effect in [21]. Consider a four-point function ⟨W ( On this plane, we can define Rindler coordinate [21] t = U sinh T, y = U cosh T (3.21) where U > 0 is for right Rindler wedge and U < 0 is for left Rindler wedge (see Fig. 8). These two wedges are related by analytic continuation T → T + iπ and the vacuum in Minkowski spacetime is equivalent to the thermal state in one of the Rindler wedge with inverse temperature 2π. We can consider each pair of W and V located in two different wedges, by which we can construct an OTOC in one Rindler wedge with inverse temperature 2π. For example, we can take Fig. 8), which leads to where the LHS is the correlation function in Minkowski vacuum and the RHS is a thermal OTOC in right Rindler wedge. The conformal Regge theory studies this fourpoint function under Regge limit with T → ∞ but fixed U . It has been shown [21] that it has a non-maximal exponential growth e λT with λ < 1. This non-maximal Lyapunov exponent comes from summing over infinitely many higher spin channels in the four-point function.
To compare with our 0+1 dimensional result, we should perform dimensional reduction by restricting to zero momentum mode along all spatial directions, which is a bit intricate. Here we will simply take U i = U for all i and compare the T -dependence with (2.122).
By conformal symmetry, the four-point function ⟨W ( where the conformal invariant cross ratios are u = The Regge limit is for T 1 , T 2 ≫ T 3 , T 4 limit, and we have [13,14], A(u, v) under this limit is given by 10 where Ω iv (ρ) is a harmonic function on (d − 1)-dimensional hyperbolic space (here by assuming U i = U we have ρ = 0), j(ν) is the leading Regge trajectory, andα(ν) is a slowly varying function of ν. In large T limit, the ν-integral can be evaluated using saddle point approximation with the saddle point given by ν = 0 [21], which gives Here C R is a constant, λ = j(0) − 1, and g W,V is the conformal correlator in Rindler wedge with no spatial separation . It is not clear whether this difference is due to we are comparing a d-dimensional theory with a (0 + 1)-dimensional system. Assuming not, we can match (3.27) with (2.122) with the identification (3.30) up to arbitrary K W,V (λ). Since this equation is essentially the same for W and V except conformal dimensions, we will suppress W, V labels in the following. Using (2.117), we have (3.31) 10 To get the correct phase e −iπ/2 , one needs to be careful about how u and v circle around origin after we set infinitesimal imaginary part T i → T i + iϵ i with ϵ 1 < ϵ 3 < ϵ 2 < ϵ 4 for OTOC. Unlike [13], both u and v circle around origin clockwise for 2π when T ≫ 0 in our case. which leads to Let us define the Fourier transformation of G(T ) as where ϵ is an infinitesimal positive number. It has been shown in [32] that From (3.33), it is clear that the Fourier transformation of G(±λ, T ) is K(±λ)e ∓iπλ/2 I(∆+ λ/2; ω ± iλ/2), which leads to From this equation and the definition (3.32), there is no unique solution for the coefficient c mn of the differential operator L T . A convenient choice is where ∂ T acts on bare operator and ∂ ϕ T acts on the effective mode. Note that this L T can be expanded in power series in both ∂ T and ∂ ϕ T . Moreover, we can expand L T near maximal chaos limit λ = 1 and find where γ is the Euler's constant. The first term is the same as (3.5) and the subleading terms can be regarded as perturbative corrections of higher spins to the vertex.

Relation to the EFT of maximal chaos
The effective field theory for maximal chaos was constructed in [11], which contains just one effective mode φ on a Keldysh contour in the thermal state with inverse temperature β = 2π. This effective mode φ has correlation function with exponential growth that explains the behavior of OTOC. In this section, we will show that our effective field theory of non-maximal chaos at maximal chaos λ = 1 can be equivalently connected to the theory in [11]. In particular, our two-component effective mode ϕ 1,2 reduces to one mode φ.
Taking the maximal chaos limit λ → 1 in (2.94), we see that both k +,± (iλ) diverge. This implies that the two operators K +,± (i∂ t ) do not contain the factor ∂ 2 t − λ 2 at maximal chaos. Physically, the two component fields η +,± (τ ; t) decouple from the dynamics of quantum chaos. Now let us examine the behavior of k −,p (iλ) in this limit.
Consider ω = iλ in (2.77); there are two equations but three parameters k −,± (iλ) and k −,0 (iλ), whose general solution is In the λ → 1 limit, with k +,0 (iλ) finite, 11 we have In the EFT of maximal chaos [11], the effective mode φ is local and only depends on one time variable. This implies that our two effective modes ϕ 1,2 at λ = 1 should not have any nontrivial dependence ont. In other words, consistency requires that η −,± must also decouple at maximal chaos, which implies From (2.100)-(2.102) we then find that in the λ → 1 limit, φ ± decouple (i.e. they are not relevant for the exponential behavior), and That is, in this limit,t-dependences drop out and ϕ 1,2 become the same field. Furthermore, from (2.103), φ has periodicity 2π in imaginary t direction, i.e. it satisfies the standard KMS condition with inverse temperature 2π. We have thus fully recovered the setup of the EFT for maximal chaos.
The EFT action (i.e. the part relevant for exponential behavior) now becomes whereφ was introduced in (2.103) and Now recall from (2.23) that only φ is relevant for the four-point function (as W and V couple to ϕ 1,2 which become φ). Furthermore, from (4.3), φ andφ decouple at ω = i. Thus for the purpose of understanding the exponential behavior of the fourpoint function, we can just keep the first term in the effective action (4.5), reducing back to [11]. As discussed in Sec. 2.1 and 2.2, the differential operator L t in the vertex that couples the bare operators and effective fields has the same form in both maximal and non-maximal cases. Moreover, in the λ → 1 limit, the shift symmetry (2.81) of ϕ 1,2 , becomes (ϕ 1 , ϕ 2 ) → (ϕ 1 , ϕ 2 ) + (e ±t , e ±t ) → φ(t) → φ(t) + e ±t . (4.7) (2.82) implies that the shift symmetry obeyed by vertex L t is which also matches with that in the EFT of maximal chaos [11].
To close this section, we note that the Wightman function G > −,0 (τ ; t) given in (2.88) diverges in the limit λ → 1. This divergence reflects that in the limit G > −,0 develops a te λt term which is not present for λ < 1. More explicitly, from (2.86) and (2.80), we find G rr −,0 (t) should satisfy whose general solution is where c 0 is an arbitrary constant. It then follows from (2.87) The presence of linear-exponential term in Wightman function at maximal chaos was already observed in [11].

Identifying the effective fields in the large q SYK model
In this section we examine the large q SYK model in some detail. In this model, fourpoint functions of fundamental fermions can be computed analytically in the Euclidean signature. We show that in this theory it is possible to identify two Euclidean effective fields ϕ E 1,2 in terms of the microscopic description, which can be identified as ϕ 1,2 (t; t) of Sec. 2 evaluated in the Euclidean section with pure imaginaryt and t. It is possible to calculate Euclidean two-point functions of ϕ E 1,2 using the microscopic description. We show that the Lorentzian analytic continuation of these two-point functions can be fully captured by the EFT of Sec. 2. This provides a stronger check on the EFT formulation than just matching the structure of OTOCs done in Sec. 3.
Two-point functions of ϵ can be obtained summing over the eigenfunctions of L [31] as follows. Eigenfunctions of L can be labeled by two quantum numbers n, m, and separated into two groups where Z + /Z − denotes even/odd integers respectively and M ± are two infinite discrete sets of real numbers with magnitude larger than 1. When n takes values in Z + (Z − ) , m takes value in M + (M − ). The two-point function of ϵ can then be written as where we define the notation The infinite sum of m in (5.17) can be evaluated in a closed form by Sommerfeld-Watson transformation [31]. For different ordering assignment for τ k , they lead to TOC or OTOC after analytic continuation τ k → it k . TOCs do not have exponential growth, while the OTOC with 2π > τ 1 > τ 3 > τ 2 > τ 4 ≥ 0 is given by After analytic continuation τ k → it k , it gives (3.1) mentioned earlier. Notice that, with τ k → it k , the cosine function in (5.17) leads to exponential growth e ±m(t 1 +t 2 −t 3 −t 4 )/2 . Since every quantum number m ∈ M ± has magnitude greater than 1, the exponential growth in each individual term in (5.17) violates the chaos bound. This infinite tower of quantum numbers m is the analogue of the infinite tower of higher spins to be summed over in the higher dimensional Regge theory.

Identifying the effective fields
We now seek an alternative way to understand two-point function of ϵ, without going through the infinite sums over m, n. We are interested only in the exponential part (after analytic continuation) of the two-point function, and would like to identify a finite number of effective fields that can capture that. For this purpose, consider general solutions to the saddle-point equation (5.4), which can be written in a form where f, g are arbitrary functions. The above parameterization of σ is not unique as the right hand side is invariant under an arbitrary SL(2, C) transformation f → a + bf c + df , g → a + bg c + dg , bc − ad = 1 .
To obtain an effective description of the exponential behavior, we first rewrite (5.22) in a more convenient form where G 0 is the equilibrium two-point function of fundamental fermions given earlier in (5.9), and ∆ = 1/q is the conformal dimension of fundamental fermionic operator ψ i . The differential operator L τ can be interpreted as a vertex that couples ϵ on−shell to χ i (τ i ), and from (5.23), it obeys the following symmetry Motived from (5.24), we write ϵ(τ 1 , τ 2 ) as whereτ = τ 1 +τ 2 2 . τ L (τ S ) is the larger (smaller) of τ 1,2 , whose usage ensures that (5.27) respects the invariance of ϵ under switch of τ 1,2 . ϕ E 1,2 are dynamical counterparts of χ 1,2 , with χ 1,2 parameterizing their classical solutions.
Equation ( Given that the action for ϕ E 1,2 must be invariant under the shift symmetry (5.23), we can be certain that correlation functions of ϕ E 1,2 must contain exponential timedependence. This establish ϕ 1,2 as the effective fields which directly captures the exponential behavior of two-point function of ϵ. 13

Two-point function of ϕ E i
In principle we can try to find the (Euclidean) action of ϕ E 1,2 by plugging (5.27) into (5.10) or (5.14), and being careful about the Jacobian in changing variables from ϵ to ϕ E 1,2 in the path integral for ϵ. It is, however, difficult to do in practice. In addition to having to understand the Jacobian, various complications discussed in Sec. 2.2-2.3, including that ϕ E 1 and ϕ E 2 are defined on different domains, should also be faced here. Here we show that using (5.14) we can nevertheless find their Euclidean two-point functions, and confirm their exponential time-dependence. In next subsection, we show that these Euclidean two-point functions can be reproduced from the EFT formulation of Sec. 2 by a suitable choice of the action there.
To compute two-point functions of ϕ E 1,2 , instead of considering (5.27) as a change of variables in the path integral, we canonically quantize (5.14) by treatingτ as "time", and treat (5.27) as an operator equation. Below we will use ϵ, ϕ E 1,2 to denote the fields in the Euclidean path integral defined with (5.14), andε,φ E 1,2 the corresponding operators in the canonical quantization. By definition, Euclidean correlation functions of ϕ E 1,2 are given by "time-ordered" correlation functions ofφ E 1,2 , i.e. (5.28) whereT denotes ordering inτ and the subscript "c.q." on the RHS is to distinguish the expectation value in Euclidean path integral on the LHS. We outline the main steps here, leaving technical calculations to Appendix D: 1. In canonical quantization of (5.14), we can expandε in terms of a complete set of modes {g m } asε (τ , x) = m g mâm + g * mâ † m (5.29) 13 Note that two-point function of ϵ has exponential behavior only when the ordering of time arguments corresponds to OTOC, while two-point functions of ϕ E always have exponential behavior.
where g m solve the equation of motion Lg m (τ , x) = 0, and obey the conditions g m (τ , 0) = g m (τ , 2π) = 0 (from (5.13)). m takes value in the same sets M ± discussed below (5.16). {g m } are assumed to be properly normalized under the Klein-Gordon inner product, such that annihilation and creation operatorsâ m andâ † m obey the standard commutation relation [â m ,â † m ′ ] = δ m,m ′ . Due to the nontrivial boundary condition (5.13) inτ direction, the system is not in the vacuum state ofâ m , and it can be shown thatâ m ,â † m have correlation functions 14 where s = ± labels two different sectors ofâ m andâ † m with m ∈ M ± respectively.
2. Since the mode functions g m solve the equation of motion of ϵ, from (5.24), they can be written as where L τ is defined in (5.25), and {χ i,m } is a complete set of basis functions for χ i in (5.24). Plugging (5.32) into (5.29), then from (5.27), we can writeφ E 1,2 aŝ Two-point functions of ϕ E i can then follow from (5.31). Notice from (5.33) that ϕ E i (τ ; τ i ) does not have anyτ dependence. So the onlyτ -dependence in twopoint functions of ϕ E 1,2 comes from θ(τ ) on the right hand side of (5.28). Such τ -dependence is precisely what we had in Sec. 2, see e.g. (2.73), except that there it came from our assumption of weakτ -dependence and derivative expansion in τ , but here for the large q SYK model it is exact.

Effective action for large q SYK
We now work out the explicit form the EFT action which match with correlation functions (5.35)-(5.36). As our EFT is formulated in Lorentzian time, we need to first analytically continue the Euclidena correlation fuctions (5.34). The correct analytic continuation for large q SYK model is ϕ E j (τ ; τ ) → ϕ j (t; t) = −iϕ E j (it; it) and L τ → L t . Note that (5.36) contains linear-exponential terms te ±λt after analytic continuation τ → it. From discussion of Appendix C, this means that we need to include quadratic order of ∂ 2 t − λ 2 in some K s,p (i∂ t ), i.e.
Indeed, there is a scaling limit in SYK model when we take maximal chaos limit. The Lyapunov exponent λ is related to the inverse temperature β and coupling J by (5.8). The maximal chaos limit corresponds to strong coupling limit J → ∞ (or equivalently low temperature limit β → ∞), under which we can solve (5.8) perturbatively as (β = 2π) However, the low temperature limit should still be understood in the regime of validity of large N , i.e. N ≫ J ≫ 1. Therefore, in maximal chaos limit, for (5.40)-(5.42) we have which by (5.43)-(5.45) implies Given the action (5.39), by (2.74) and (2.75) the exponential terms in retarded correlation functions scale as where c 1,2 are two O(1) numbers. From (5.48) and (5.49) in maximal chaos, there is an enhancement of J to ∆ s,0 (t) while ∆ s,± (t) are still O(1). This means that η s,± decouple at maximal chaos at leading order of J . Sinceτ dependence only exists for p = ±, this means that at leading order of J two effective modes ϕ 1,2 reduce to a single field at maximal chaos following the same argument for (4.4). Note that the first term of ∆ −,0 (t) in (5.50) is O(J ) but the second term is O(1). Therefore, at leading order of J , ∆ −,0 (t) only has pure exponential terms. Using the explicit expression for ∆ s,p (t) in (C.25)-(C.28), one can show that It follows that K −,0 (i∂ t ) in the effective action (5.39) can be reduced to be just linear in ∂ 2 t − λ 2 and we can take ansatz This is the same condition we impose in (4.3) for maximal chaos. Following the discussion in Section 4, this EFT reduces back to case in [11]. Note that the overall scaling N/J in (5.53) reflects the fact that the effective action for SYK model in strong coupling/low temperature limit is proportional to N/(βJ ), which is observed in the Schwarzian action [29].

Higher order terms and exponentiation
Higher order terms in equation (1.3) are suppressed by higher powers of 1/N . Here we show that a subset of terms of the form e kλt N k (k an integer) can be rensummed and exponentiated. Such terms dominate in the regime N → ∞ and t ∼ 1 λ log N such that e λt N is finte. These contributions come from including higher powers of effective fields ϕ i in the product (2.19), but in the effective action still keep only quadratic terms. 15 The full four-point functions of V and W then involve multi-point correlation functions of effective fields ϕ i , which factorize to products of two-point functions. We show that the shift symmetry on the vertices that couple W (t 1 )W (t 2 ) to higher powers of ϕ i implies that TOCs again do not have exponential growth, and the OTOC (recall (2.105)) has the exponentiated form 15 Including nonlinear terms in the EFT action leads to higher order terms of the form e k 1 λt N k 2 with k 2 > k 1 .
where various notations will be explained below. We now proceed to describe the derivation of (6.1).

Towards a scattering formula
Without derivative ont, all order generalization of the vertex can be written as where in the first line ⟨·⟩ 0 means taking the expectation value of bare operators, and we assume ℑt 1 < ℑt 2 so that the arguments for ϕ i is t i (i = 1, 2); and in the second line we defined the notation where C is a tensor function of t 12 . Note that separate t 1,2 derivatives on W 0 (t 1,2 ) in (2.20) and (2.21) are combined to act on the argument of g W (t 12 ) by translation symmetry. In (6.2), the range of the sum for m is from 0 to ∞ but the sum for j, k s could be either finite or infinite for each m. For simplicity, we will consider their ranges to be finite at each m. For each k s , we choose the range of sum to be the same, i.e. k 1 , · · · , k m ∈ {0, · · · , d m }. This choice defines ϕ I in (6.4) as a 2d m dimensional vector at each m. It follows that C I 1 ···Im can be chosen as a symmetric tensor of order m by the permutation symmetry of among all ϕ I in (6.3). For m = 0, we have C j = δ j,0 because the leading order of (6.2) is just g W (t 12 ). Similarly expansion applies to ⟨V (t 3 )V (t 4 )⟩ for ℑt 34 ∈ (−2π, 0) with notation With four time variables (t 1 , t 2 , t 3 , t 4 ) in the analytic domain D, a generic four-point functionF iŝ With the effective action being quadratic, the (m + m ′ )-point function of ϕ i factorizes into products of two-point functions. We will ignore the self-interaction terms (those pairs of ϕ i with samet) because they do not grow in OTOC. It follows that m ′ = m andF where the the permutation symmetry in Wick contraction gives m! that cancels one m! in the denominator. The KMS condition requiresF invariant under t 1 → t 2 − 2πi and t 2 → t 1 (and also t 3 → t 4 − 2πi and t 4 → t 3 ). Using (2.26)-(2.27), we find coefficients C andC must satisfy the constraints where the map i →ī means 1 ↔ 2. This is equivalent to We further require the right hand side of (6.3) to be invariant under shift symmetry (2.81), which lead to C I 1 ···Im (∂ k 1 δϕ i 1 ) · · · ϕ Im = 0, δϕ i ≡ (δϕ 1 , δϕ 2 ) = (e ±λ(t 1 +iπ) , −e ±λt 2 ) (6.11) for arbitrary ϕ I . This is a quite strong constraint. Let us define the 2d m dimensional vector e I = ∂ k δϕ i at each m and (6.11) becomes I 1 C I 1 ···Im e I 1 = 0 (6.12) At each m, for the symmetric tensor C we can always find a linear independent set of vectors {u (m)a } such that where D m is the size of the set, and ξ and used (6.15) in the last step for both u (m)a ± andũ (m)a ± . Since we are only interested in large Lorentzian time separation t 1 , t 2 ≫ t 3 , t 4 or t 1 , t 2 ≪ t 3 , t 4 , we only need to keep the exponentially growth term in (6.18) because the other term is exponentially suppressed. Let us take t 1 , t 2 ≫ t 3 , t 4 , then (6.17) becomes with the exponential growth Assume we can do an inverse Mellin transformation to define h andh as Then we can rewrite (6.20) as This has exactly the same form as [19,20], and also matches with the trans-plankian string scattering formula near horizon of a AdS black hole [4]. In [19], this scattering formula was conjectured by a heuristic argument and the authors of [20] later proved it with a specific structure of Feynmann diagrams. In this work, we show that (6.24) holds in a more general scenario since it is just a result of a shift symmetry. Note that the assumption of inverse Mellin transformation requires analyticity in m. This is a nontrivial constraint on the vertices because at each level of m one could have choosen ξ (m) a and the sets D m andD m quite randomly in a pattern without analyticity in m. However, there is a simple and sufficient way to guarantee analyticity, which requires three parts: 1. The total ways of coupling between bare operator and effective mode does not change as we increase the number m of effective modes, which picks D m andD m as some fixed sets for all m, in which the vector dimension 2d m is also fixed.
2. All these types of couplings exist at any level of m, which releases the m dependence of u (m)a + .

The coefficient ξ
is an analytic function of m.
It is noteworthy that the ordinary ladder diagrams in eikonal scattering obeys these three conditions. Therefore, this explains why the exponentiation in (6.24) is also a result of eikonal scattering [4], in which e −Xyỹ is the eikonal scattering amplitude, h is the wave function of two W 's andh is the wave function of two V 's.

An example
In this subsection, we will present a simple example following the general construction in the last subsection. We will solve a simple orthogonal vector u I , find the vertices leading to this vector u I , compute the exponentiation formula and compare it with the known result of large q SYK model [20]. Let us first solve the orthogonal vectors u a I . Define q = e −λ(t 12 +iπ) and the equation u a I e I = 0 can be written as Without loss of generality, we can assume u a I is a polynomial in q We can choose the normalization of u I such that it is invariant under KMS transformation q → 1/q and i →ī. In this case, by the discussion below (6.16), we can construct a KMS invariant C just using this vector. It is easy to see that the appropriate normalization is q −1/2 and the orthogonal vector is Then we will solve the vertices with coefficients C i 1 ···im j;k 1 ···km leading to this vector u I for each order m. Let us start with m = 1. We can further impose a simple condition that j only takes values 0 and 1 in (6.4). Define n-th order ∂ t 12 derivative to bare correlator g W (t 12 ) = g(q) as g n (q) = (−λq∂ q ) n g(q) with g 0 (q) = g(q). For c ̸ = 0, ∞, comparing (6.4) and (6.13) leads to Our goal is to solve coefficients C i j;k . However, for arbitrary C i j;k and ξ ( 1)(q), this is also a differential equation for g(q), which will constrain g(q) to a specific form. We solve (6.31) in detail in Appendix E and present the result here. The correlation g(q) must be in the form g(q) = (q 1/2 + q −1/2 ) −2∆ (6.32) up to normalization and with a constant ∆. It is obvious that this g(q) obyes KMS symmetry g(q) = g(1/q). With this solution, the coefficients C i j;k are where we choose the normalization such that C 1 1;0 = 1. One can check that this solution obeys KMS symmetry (6.9). Taking them into (6.31) and using (6.30), we find which is explicitly KMS invariant as expected.
In the two component vertex form, we have where on the RHS we have suppressed the notation and the derivatives only act on the second argument of ϕ i (t; t i ). Note that the first line, namely c = 1, exactly matches with the vertex (3.5) of large q SYK model. To construct the higher order coupling coefficient C i 1 ···im j;k 1 ···km such that (6.13) holds for all m is not hard for this example. The point is to note the following feature g n (q) g(q) = P n (q) (1 + q) n (6.37) where P n (q) is a n-order polynomial has no factor of (1 + q). Therefore, for any n-order polynomial P n (q), we can pick a linear combination such that n j=0 w j g j (q) g(q) = 1 (1 + q) n n j=0 w j (1 + q) n−j P j (q) = P n (q) (1 + q) n (6.38) Since q 1/2 u I (q) is a first order polynomial of q, the product q m/2 u I 1 (q) · · · u Im (q) is a polynomial of q up to order m for each choice of (I 1 , · · · , I m ). It follows that we can choose C i 1 ,··· ,i k j,k 1 ···km such that where z(m) is a constant that could depend on m. It clear that (6.39) obeys KMS symmetry (6.10).
If we only include this type of orthogonal vector u I in C, we have where we assume the vertex of V consists of the same vector u I but with a possibly different coefficientz(m). For analytic functions z(m) andz(m) that decay fast enough along ℑm → ±∞, the Mellin inversion theorem guarantees the existence of h(t 12 , y) andh(t 34 ,ỹ). However, to determine the exact form of z(m) andz(m) needs detailed knowledge of the dynamics of underlying UV model (for example [20]). For large q SYK model, we can choose W = V to be the fundamental Majorana fermion ψ, whose conformal weight is ∆ = 1/q. The α parameter in (6.21) is given by (3.5). Its exponentiation exactly falls into above case of c = 1 with the following choices of z(m) andz(m) where g ψ (t) is the fermion correlation function (3.2). One can check that this exactly matches with the result in [20].

Conclusion and discussion
In this paper, we constructed an effective field theory to capture the behavior of OTOCs of non-maximal quantum chaotic systems. While the theory is constructed phenomenologically, we showed that it is constraining enough to predict the general structure of OTOCs both at leading order in the 1/N expansion, and after resuming over an infinite number of higher order corrections. These general results agree with those preciously explicitly obtained in specific models. We also showed that the general structure of the EFT can in fact be extracted from the large q SYK model, providing further support for its validity. There are many future directions to explore, on which we make some brief comments.
Higher dimensional systems A most immediate direction is to generalize the current discussion to higher dimensional systems. Including spatial directions will make it possible to consider much wider range of physical issues, for example, operator growths and scrambling in spatial directions, the behavior of the butterfly velocity [3,21,33,34], connections between quantum chaos and energy as well as charge transports [8,11,35], and so on. In the case of maximal chaos, a phenomenon that connects chaos and energy transport is the so-called pole-skipping [11,36]. Understanding what happens to this phenomenon for non-maximal chaotic systems is of interests. In [37] it was conjectured that pole-skipping survives in non-maximal system and the location of pole-skipping is given by B is an upper bound of the true butterfly velocity u B . An EFT including spatial directions could help check the conjecture and understand connections between energy transport and chaos in more general systems.
Physical nature of the effective fields and the shift symmetry Here we introduced chaos effective fields and shift symmetry on phenomenology ground. In the example of the large-q SYK model, we can identify the effective fields and origin of shift symmetry from the microscopic system. It is, however, not clear whether the understanding obtained in this model can be applied to general systems.
In maximal chaotic holographic systems, the shift symmetry of the EFT should be related to the existence of a sharp horizon. It is an outstanding question regarding the nature of the horizon when including stringy corrections on the gravity side, understanding the physical nature and origin of the shift symmetry for non-maximal case could provide hints for this question.
Effective field theories for Reggeons As mentioned in the Introduction (see Fig. 1b), there is a close connection between the exponential behavior in non-maximally chaotic systems and scattering amplitudes in the Regge limit. Our formulation of an EFT for non-maximally chaotic systems could provide new ideas for formulating effective field theories for Reggeons. More explicitly, the stringy scattering processes corresponding to OTOCs in holographic systems can be described by the BFKL Pomeron [4,38]. The effective fields we identified could shed light on an effective description of the Pomeron.

A A few oversimplified constructions
In this appendix, we list two oversimplified constructions of EFT for non-maximal chaos, which are slightly generalized from the EFT for maximal chaos [11,12] with one time argument. It turns out that both constructions are only compatible with maximal chaos. The purpose of this section serves as a support for the construction in Section 2 as a minimal and sufficient generalization to account for non-maximal chaos. In particular, including two time arguments in the effective modes is necessary.

A.1 Multiple effective modes with one time argument
The simplest generalization of [11,12] is to include more effective modes but still formulated with one time argument. Let us label these effective modes as φ µ with µ = 1, · · · , D for some finite D. The four-point functionF W W V V (t 1 , t 2 ; t 3 , t 4 ) is symmetric under exchange of t 1 ↔ t 2 and t 3 ↔ t 4 respectively. Therefore, to define the coupling between a bare operator W 0 and effective modes φ µ , we must respect this symmetry. There are two simple ways: one is that every W 0 couples with all φ µ , the other is coupling in an ordered way (just like (2.20) and (2.21) for two effective modes). Here we will consider the first choice.
In this case, the coupling in linear order of φ µ is where µ is summed from 1 to D and L µ t is a set of differential operators andL µ t is defined similarly with c µ nm →c µ nm . To quadratic order of φ µ , the four-point function isF where g W,V are short for g W (t 12 ) and g V (t 34 ) and ⟨T φ µ φ ν ⟩ is the Euclidean time ordered two-point function in the thermal state. Imposing the same shift symmetry [11] φ µ (t) → φ µ (t) + αe ±λt (A. 4) in the effective action, we will have the following exponential terms in the Wightman function ⟨φ µ (t)φ ν (0)⟩ = d µν e λt +d µν e −λt (A.5) where c µν andc µν are two nonzero constant matrices. Let us consider OTOC F 4 and TOC G 4 defined as where t 1 , t 2 ≫ t 3 , t 4 or t 1 , t 2 ≪ t 3 , t 4 . For TOC, we have where b.c. means bar-conjugate which replaces c µν withc µν and swaps e λt ↔ e −λt . For OTOC, we have If both sides of (A.15) are zero, we have Similarly, using the first equation of (A.14), we will have either λ = 1 or Then we can consider another TOC H ′ 4 = ⟨V (t 3 )V (t 4 )W (t 1 )W (t 2 )⟩. Following a similar analysis for H 4 (which simply swaps t 1,2 ↔ t 3,4 and W ↔ V everywhere), we will have either λ = 1 or Taking both (A.17) and (A.19) into (A.8), we find that the exponential terms proportional to d µν vanish in F 4 . Similarly, we can show that the bar-conjugate terms vanish as well. This means that if we require exponential growth of OTOC but no exponential growth in TOC, we must have maximal chaos λ = 1 in this model.

A.2 Two effective modes with one time argument and ordered coupling
As we mentioned before, for two effective modes φ i with i = 1, 2, there is another simple way to couple bare operators with them in a symmetric way. This is essentially the same as our proposal (2.20) and (2.21) in which W 0 (t S ) couples with φ 1 (τ S ) and W 0 (t L ) couples with φ 2 (t L ) where t L,S is the one of t 1 and t 2 with larger (smaller) imaginary part. The only difference is that here we will consider the effective modes with one time argument. In other words, there is not argument. For this model, we can simply substitute ϕ i (t, t) in Section 2.2 with In particular, the KMS symmetry (2.26)-(2.27) reduces to This means that the two effective modes φ i (t) are completely degenerate to a single effective mode φ(t) in a thermal state with inverse temperature 2π. It reduces to the case in [11,12] which is inevitably restricted to the maximal chaos.

B Unitary and dynamical KMS conditions
The periodicity (2.45) along imaginary time direction can be understood as η s,p in a thermal state with imaginary chemical potential. Let as define the charge carried by η s,p (τ ; t) as Q = p, namely It follows that the state consistent with the KMS condition (2.44) is whereŜ is the operator that takes the s value of η s,p , namelŷ For the state given by (B.2), the time reversal transformation T of ρ is given by where we assumed time reversal invariance ofŜ, Q and H (and also hermicity ofŜ and Q). The time reversal transformation T of η s,p (τ ; t) is defined as flipping t andτ simultaneously From this definition, T 2 = 1. In this definition, the periodicity of η s,p (τ ; t) alongτ direction is consistent  This means that the effective action does not have K r···r η r · · · η r term. In other words, each term in the action must contain at least one η a s,p .
2. Taking complex conjugate of (B.7), assuming J * s,p = J s,−p , it is clear that To guarantee this condition, we need to impose T e −i´J (2) s,−p (τ ;t)η (2) s,−p (−τ ;−t) = Tr Te i´J (1) s,−p (τ ;t)η (1) s,−p (−τ ;−t+iπ)se 2πip/3 ρ T e −i´J (2) s,−p (τ ;t)η (2) s,−p (−τ ;−t) = Tr ρ T e −i´J (2) s,p (τ ;t)η (2) s,p (−τ ;−t) Te i´J (1) s,p (τ ;t)η In terms of a-r fields, we can rewrite (B.17) as where we defined two operators  which holds for any local, time t translational invariant and (bothŜ and Q) charge conserved action S. In other words, each term in the action should be in the form of The second case we need to check is the consistency between (B.14) and (B.28). It is easy to see that they together lead to another inequality to S for α 1 , · · · , α k contain odd numbers of a. Let us focus on the quadratic action, it is given by (2.60), which is copied here In non-dissipation case, we have K aa = 0, which implies Taking this back to (B.34), one can easily see that S[η r s,p , η a s,p ] factorizes as is a real action.
C Generalization to polynomial-exponential case The differential operator K s,p (i∂ t ) can be generalized to contain higher powers of ∂ 2 t −λ 2 , which would lead to polynomial-exponential behaviors t k e ±λt in correlation functions. However, it turns out that most of these polynomial-exponential terms are excluded by the self-consistency condition (2.76): there are two cases ℑt W < ℑt V and ℑt W > ℑt V in the TOC G 4 , which must smoothly match each other at ℑt W = ℑt V .
If we only have pure exponential terms, it is automatically compatible with this condition because G 4 does not grow exponentially in both cases due to G rr + = 0 (up to non-exponential terms) as explained in Section 2.8. However, as we show in this section that this matching condition of G 4 becomes nontrivial when we include polynomialexponential terms. It turns out that only pure exponential and linear-exponential terms are allowed in correlation functions, which implies that K s,p (i∂ t ) can at most contain quadratic ∂ 2 t − λ 2 . Furthermore, we will solve the most general Wightman functions of effective modes that obey all three constraints (2.56), (2.76) and (2.79).

C.1 Wightman functions
It is convenient to work directly with Wightman functions. Let us assume for p = 0, ± that where each h s,p (t) contains polynomial-exponential pieces where n is finite number and the coefficient choice manifests KMS condition (2.48). It follows that The smoothness condition (2.56) can be written in this case as Let us explicitly implement the requirement that G 4 is smoothly defined at ℑt W = ℑt V . For ℑt W < ℑt V , G 4 is given by (2.108). For ℑt W > ℑt V , G 4 is given by flipping the sign of the first argument of G > s,p , namely The reason that G rr + = 0 leads to G 4 = 0 in (2.108) is that the shift symmetry of vertex (2.82) transforms the four pure exponential terms in (2.108) to just one exponential function of t 2 and t 4 in (2.109). However, for polynomial-exponential terms, we do not have an extended symmetry transforming, say, t k 1 e λt 1 to t k 2 e λt 2 . Therefore, to make sure (2.108) matches with (C.6) at ℑt W = ℑt V for the quadratic and higher polynomial-exponential pieces, we must require the four terms in both (2.108) and (C.6) match with each other separately where "≃" means that the equation hold for t k e ±λt terms with k > 1. The k = 1 case is a little bit different and will be discussed later. It follows that s h s,p (t) ≃ 0, p = ± (C. 8) where there is no constraint to p = 0 piece simply because G > s,0 (t) is independent onτ . Using ansatz (C.2), we can easily show order by order from (C.8) for p = ± that γ k s,± = 0, k = 2, · · · , n (C.9) Then by G rr + = 0 (2.93), we have h +,0 (t) + h +,0 (−t) ≃ 0 =⇒ γ k +,0 = 0, k = 2, · · · , n (C.10) On the other hand, using (C.1), (C.4), (C.5) and (C.10) that In this way, the consistency conditions and shift symmetry kill all quadratic and higher polynomial-exponential terms.
For the linear-exponential term, the shift symmetry of vertex (2.82) does help a bit because, for example, However, this is still not enough to save nontrivial linear-exponential term in h s,± (t).
With the existence of linear-exponential growth term, the constraint to the pure exponential term is slightly different. Comparing the pure exponential piece in (C.2) and (2.112), we should identify Taking this into (2.93) and (C.5) leads to γ 0 +,± = − cos (πλ/2) cos π(λ/2 ∓ 1/3) γ 0 Taking these solutions into (2.110) and (2.111), we will find both C 1 and C 2 vanish. One can also check (C.6) vanishes as well. This confirms the result in Section 2.8 that G rr + = 0 and smoothness condition (C.5) imply vanishing pure exponential terms in TOC for both ordering of ℑt W,V .

C.2 The effective action
The above most general consistent Wightman functions correspond to the effective actions in the form of (2.69) with where we see linear-exponential terms appear because of the double poles in 1/K −,p (ω) at ω = ±iλ.
Take ansatz (C.2) with n = 0 for s = + and n = 1 for s = −. Using (C.4) and comparing with (C.25)-(C.28), we find and other γ k s,p are given by (C.18)-(C.20). By the conclusion from last subsection, this implies that the action with (C.21) and three constraints (C.22)-(C.24) lead to absence of exponential terms in TOC.

D Correlation functions of effective modes in the large q SYK model
In this appendix, we first ignore the prefector N/(8q 2 ) in the action (5.14). In the end, this prefactor simply adds a 8/(N ∆ 2 ) factor to any two-point function.

D.1 Canonical quantization trick
As explained in Section 5.3, we will take the mathematical trick of canonical quantization to solve the Euclidean two-point function of ϵ(τ , x). First, we need to solve the equations of motion Lg m (τ , x) = 0 for wave function g m (τ , x) with UV condition g m (τ , 0) = g m (τ , 2π) = 0 from (5.13) and expand the quantized fieldε(τ , x) in terms of whereâ m andâ † m are annihilation and creation operators obeying canonical commutation relation [â m ,â † m ′ ] = δ m,m ′ , and g m is well-normalized under Klein-Gordon inner product, which is defined in (5.30). Note that the quantum number m must be discrete because the spatial direction x is finite. Due to translation symmetry inτ , we can choose the positive energy wave function for which the Hamiltonian H has eigen value m and Since g m (τ , x) is on-shell, following from (5.22) it can be expanded as whereτ in χ i,m (τ ; τ i ) is just a dummy argument whose dependence is trivial. It follows that we can rewrite (D.1) aŝ where L τ is defined in (5.25). By the fundamental domain D ϵ , the defining domain for Note that the path integral is defined on the finite spacetime D ϵ . To compute any correlation function of quantized fieldε on D ϵ , we must first specify the states on time sliceτ = 0, π respectively. This is dictated by the boundary condition of ϵ(τ , x) in the Euclidean path integral on D ϵ atτ = 0, π. From the first equation of (5.13), we see that any configurations of ϵ(0, x) is identified with ϵ(π, 2π − x). For our canonical quantization trick, this implies that we need to trace over all states atτ = 0, π with a reflection x → 2π − x. More explicitly, we consider the following Wightman function where the time evolution forτ = π is present and P is the reflection operator Therefore, we will define the expectation ⟨· · · ⟩ c.q. as ⟨· · · ⟩ c.q. ≡ 1 Z Tr P e −iπH · · · (D.9) Expanding inâ m andâ † m , we have (D.10) Since the Euclidean two-point function ⟨ϵ(τ , x)ϵ(τ ′ , x ′ )⟩ is symmetric under exchange ofτ , x withτ ′ , x ′ , this corresponds to the Feynman propagator ofε(τ , x), i.e.
whereT is the time ordering ofτ andτ −τ ′ is restricted to [−π, π]. We emphasize that the LHS of (D.11) is a Euclidean two-point function and the RHS of (D.11) is just a mathematical trick to compute it in terms of a Feynman propagator of quantized field ϵ by regardingτ as "time". The quantized fieldε and the corresponding Hilbert space are part of the trick and have no physical meaning. On the domain D ϵ , taking above equations into (5.11) leads to where one should note that the dummy argumentτ plays a role in the time orderinḡ T though its explicit dependence in (D.4) is trivial. Given (D.5), it is noteworthy that the reflection (D.8) acts nontrivially onχ i . The transformation (τ , x) → (τ , 2π − x) is equivalent to (τ 1 , τ 2 ) → (τ 2 + π, τ 1 − π), which implies Pχ 1 (τ , τ )P −1 =χ 2 (τ , τ − π), Pχ 2 (τ , τ )P −1 =χ 1 (τ , τ + π) (D.14) Taking this back to (D.13) and using definition (D.9), we have the following KMS conditions 16) where τ ∈ [0, π] by the defining domain ofχ i . It is very interesting that the Euclidean four-point function (D.13) now has the same structure as (2.23). Therefore, we would like to define the Euclidean two-point function of two effective fields ϕ E 1,2 (τ ; τ ) such that Take this definition into (D.13) and analytically continueτ k → it k . Comparing with (2.23), we will identify the Euclidean fields ϕ E i as ϕ i after analytic continuation. In particular, the two KMS conditions (D.15) and (D.16) in terms of ϕ E i are equivalent to the KMS conditions of ϕ i (2.26)-(2.27). and for τ 1 = τ 2 + 2π (x = 2π) leads to

D.4 The effective action
The effective action will be still formulated in η s,p (τ ; t) variables and the correlation functions of ϕ i will be given by (2.49) to (2.51). Since the correlation function (D.54) contains linear-exponential terms, the effective action needs to contain quadratic factor of ∂ 2 t − λ 2 . It turns out that we need to take the action (2.69) with K +,p (i∂ t ) = (∂ 2 t − λ 2 )k +,p (i∂ t ), K −,p (i∂ t ) = (∂ 2 t − λ 2 ) 2 k −,p (i∂ t ) (D.58) where K −,p (x) has a double zero at ±iλ. By symmetry (2.68), we have k s,p (x) = (−) p k s,−p (−x). This effective action has been thoroughly discussed in Appendix C.2.
To match the correlation functions (D.52) with this EFT action, let us first compute T ϕ i (t; t)ϕ j (0; 0) correlation functions by (2.49)-(2.51) with the most general consistent Wightman functions solved in Appendix C.1, which are also results of the action (D.58) by the analysis in Appendix C.2. Then we compare T ϕ i (t; t)ϕ j (0; 0) with (D.52) after analytic continuation. Since these correlation functions obey the same KMS conditions, matching one of them is sufficient. In the following, we will take i = j = 1 and assume ℑt ∈