Quantum Correction to Chaos in Schwarzian Theory

We discuss the quantum correction to chaos in the Schwarzian theory. We carry out the semi-classical analysis of the Schwarzian theory to study Feynman diagrams of the Schwarzian soft mode. We evaluate the contribution of the soft mode to the out-of-time-order correlator up to order $\mathcal{O}(g^4)$. We show that the quantum correction of order $\mathcal{O}(g^4)$ by the soft mode decreases the maximum Lyapunov exponent $2\pi/ \beta$.


Introduction
Recently, quantum chaos has been intensively investigated in AdS/CFT. One of the diagnosis of the chaos is the well-known butterfly effect which stands for the sensitivity of a system on the initial condition. In quantum system, such sensitivity can be captured by the out-oftime-ordered correlator (OTOC) defined by [1][2][3][4][5] V (t)W (0)V (t)W (0) β , (1.1) where · · · β denotes the thermal expectation value at temperature β −1 . The sensitivity of a chaotic system on the initial condition leads to an exponential growth of the OTOC between the dissipation time t d ∼ β and the scrambling time t * [1][2][3][4][5]: Here, g 2 is proportional to the inverse of the large central charge c, and κ is a constant depending on the details of models. Note that the scrambling time t * is of order 1 λ L log 1 g 2 . The exponential growth rate λ L is known as the Lyapunov exponent. It was shown [5] that the Lyapunov exponent λ L is bounded in a quantum field theory with unitarity and causality. 1 i.e., This bound on chaos indicates the concept of the maximal chaos, and it was shown that SYK models [7][8][9][10][11][12][13][14][15][16][17][18][19], the tensor models [16,[20][21][22][23][24], the dilaton gravity on nearly-AdS 2 [25] and string worldsheet theories [26][27][28] have the maximal Lyapunov exponent λ L = 2π β . In such a maximally chaotic system, (1.2) is almost constant in early time because the exponential growth is negligible compared to the leading constant term. As time is increased, the subleading exponential growth with λ L = 2π β in (1.2) is comparable to the leading constant around the scrambling time t * ∼ 1 λ L log 1 g 2 . At the same time, the next sub-leading term of order O(g 4 ) would also become of importance. Then, it is interesting to ask a question whether the next sub-leading correction of order O(g 4 ) increases or decreases the maximum Lyapunov exponent λ L = 2π β . In this paper, we will make an attempt to answer this question 2 in the Schwarzian theory [35][36][37][38][39][40][41] which describes the low-energy sector of the SYK-like models and the dilaton gravity on the nearly-AdS 2 . And, it is responsible for the saturation of the bound on chaos thereof. Although not all the quantum correction of order O(g 4 ) ∼ O( 1 c 2 ) to the OTOC exhibits universality, the contribution of the Schwarzian modes of order O(g 4 ) will be universal.
The outline of this paper is as follows. In Section 2, we review the semi-classical analysis of the Schwarzian theory. Then, we evaluate the propagator of the Schwarzian soft mode and its loop correction. In Section 3, we consider a bi-local field and its Euclidean two point function which corresponds to four point function of a fundamental local field. By studying the soft mode expansion of the dressed bi-locals, we evaluate the contribution of the soft mode to the Euclidean two point function of bi-locals. Then, taking analytic continuation to the real time, we obtain the soft mode contribution to OTOC. In Section 4, we make concluding remarks, and we present caveats and future directions.

Review
We begin with the review of the Schwarzian theory in [35]. The partition function of the Schwarzian theory is given by where φ(τ ) ∈ Diff(S 1 ) is a diffeomorphism of a circle. The Schwarzian theory has SL(2, R) symmetry given by and we mod out the SL(2, R) volume. Hence, the physical degree of freedom of the Schwarzian theory lives on the quotient space Diff(S 1 )/SL (2). Note that µ[φ] is the reparametrization invariant measure. After SL(2, R) gauge-fixing, the measure becomes [35,42,43] The measure µ[φ] can be exponentiated by introducing a fermion ψ(τ ), and the partition function of the Schwarzian in (2.1) can be written as [35] Z where the action S is given by In the weak coupling limit g 1, one can perform the semi-classical analysis of the action in (2.5) by expanding the diffeomorphism φ(τ ) around a saddle point φ(τ ) = τ : Accordingly, the Schwarzian action in (2.5) can be expanded with respect to g: where we have After fixing the SL(2) gauge [35] dτ (τ ) =ˆdτ e ±iτ (τ ) = 0 ,ˆdτ ψ(τ ) =ˆdτ e ±iτ ψ(τ ) = 0 , (2.11) we Fourier-transform the soft mode (τ ) and the fermion ψ(τ ) to the (discrete) momentum space:   They give the free propagator as well as cubic, quartic vertices of the soft modes and fermions (see Figure 1 ∼ 2). For the leading quantum correction to OTOC, it is enough to consider the interactions up to quartic vertex.

Soft Mode Propagator
From the quadratic action in (2.13), one can read off the free propagator of the soft mode and the fermion in (discrete) momentum space: .
Now, we will evaluate the loop correction of order O(g 2 ) to the propagators in (2.16). At order O(g 2 ), there are two types of loops: one with a quartic vertex and one with two cubic vertices. And, each loop correction gives divergence. However, as in the calculation of the free energy [35], the divergence in the bosonic loop is cancelled with that of the fermion loop of the same type, which leads to a finite answer.
One-loop with One Quartic Vertex: We evaluate the contribution of the two diagrams in Figure 3 with bosonic and fermi loops made of quartic vertex: . (2.17) Note that each series does not converge because the term of each series does not converge to zero as m goes to infinity. However, the summation of two terms converges to zero as m → ∞, and its series can be expressed as contour integral as follow. .
where the contour C is a collection of small counterclockwise circles centered at ζ ∈ Z/{−1, 0, 1}. By deforming the contour, it can be changed into the sum of the residues at ζ = −1, 0, 1:  One-loop with Two Cubic Vertices: In a similar way, one can calculate the sum of two diagrams with bosonic and fermi loop composed of two cubic vertices in Figure 4. Again, each diagram does not converge, but the summation of the two diagram gives a finite contribution. .
It can also be written as contour integral as before. .

Out-of-time-ordered Correlator
We will evaluate the contribution of the Schwarzian soft mode to OTOC. For this, we first evaluate Euclidean four point function in a specific configuration, and then we will take analytic continuation to real time OTOC.

Dressed Bi-local Field
The OTOC is basically a four point function of "matter" fields. In the SYK model, the OTOC of the fundamental fermion χ i (τ ) (i = 1, 2, · · · , N ) was evaluated [12,[16][17][18]44]. This can also be viewed as two point function of the bi-local field ψ(τ 1 , . A similar bi-local field from a matter scalar field was used to evaluated OTOC in the two-dimensional dilaton gravity on the nearly-AdS 2 [25]. Here, the gravity sector is described by Schwarzian action, and the scalar matter field was included on top of the Schwarzian mode. On the other hand, without coupling to an extra matter field, one can also construct the bi-local field by a (boundary-to-boundary) SL(2) Wilson line of BF theory for AdS 2 [39,40,[45][46][47] or Chern-Simons gravity for AdS 3 [48,49]. 3 One can consider a smooth fluctuation around the constant background (e.g., BTZ black hole) with a fixed holonomy along the time circle. This fluctuation can be described by Schwarzian action on the boundary [41,48,49,52]. For OTOC, one can study Wilson line evaluated with the SL(2) gauge field corresponding to the smoothly fluctuated background, which can be interpreted as bi-local field dressed by the soft mode [48,49]. Note that this soft mode generates conformal transformation on the boundary. Hence, the dressed bi-local field can equivalently be obtained by the conformal transformation of the (boundary-to-boundary) two point function. Then, one can expand the dressed bi-local field with respect to the soft mode: where h is the conformal dimension of the matter field, and Φ cl (τ 1 , τ 2 ) is the leading term in the soft mode expansion which corresponds to the two point function in the constant background. Here, we defined the center of time χ and the relative time σ to be In addition, the soft mode eigenfunction f (1) n (σ) and f (2) m,n (σ) in (3.1) are found to be In particular, it is convenient to evaluate them at σ = − π 2 : The form of f m,n,p (σ) is complicated, but it is enough to evaluate it at a particular value for our purpose. (3.7)

Euclidean Four Point Function
In this section, we will evaluate the contribution of the soft mode to the Euclidean two point function of the dressed bi-local fields Φ dressed , which corresponds to four point function of a matter field: The leading contribution is the product of the one point function of the bi-locals which corresponds to the disconnected diagrams in the four point function of the matters in the constant background.
where the leading one-point function Φ cl (τ 1 , τ 2 ) is given by Note that one can evaluate the correction to the one point function of the bi-local field. For this, we perform the soft mode expansion of the one point function: −n,n (σ) + · · · =1 + 1 4 gh(4 + 8h − π 2 ) + · · · . (3.11) (e) Soft modes dressing ∼ O(g 4 ) Figure 5: Diagrams for the contributions of the soft mode to the Euclidean two point function of bi-local fields. We represent the dressed bi-local field by a dot.
Note that Φ dressed 0 is independent of χ because of SL(2) charge. In this paper, we will consider a fixed value of σ (i.e., σ = − π 2 ), and then Φ dressed 0 is nothing but a numerical constant. One may evaluate the Euclidean four point function for any choice of (τ 1 , τ 2 ; τ 3 , τ 4 ). However, for the analytic continuation to a particular OTOC, it is enough to consider the following configuration which simplifies the calculation of the four point function: where χ ∈ (−π/2, π/2). Therefore, we will evaluate the following Euclidean four point function.  Figure 5a) is found to be where we used the leading propagator of the soft mode in (2.16). This can be written as the following contour integral .

Real-time Out-of-time-ordered Correlator
From the Euclidean four point function in (3.25), we take the analytic continuation from Euclidean time χ to Lorentzian time t (See Figure 8) (3.26) in order to obtain the OTOC: where the ellipsis represents terms that do not grow exponentially at order O(g 4 ).
First of all, note that the leading Lyapunov exponent saturates the bound on chaos [6,12,25]. i.e., (3.28) Among the contribution of order O(g 4 ) in (3.27), a term like − 2h 3 β g 4 te 2πt β has been observed to give a correction to the maximum Lyapunov exponent. In SYK model, for instance, the contribution of the non-zero mode to OTOC gives te 2πt β which leads to the 1/βJ correction to the leading Lyapunov exponent [12]. While the 1/βJ correction in SYK model decreases the Lyapunov exponent from the maximum value 2π β , the 1/c correction to the Lyapunov exponent from the Virasoro conformal block in large c was shown [29,30] to increase the Lyapunov exponent. i.e., λ L = 2π β (1 + 12 c ). In our result, the contribution − 2h 3 β g 4 te 2πt β in (3.27) seemingly plays a role of increasing the Lyapunov exponent. However, we have other terms at order O(g 4 ) which grow faster than te 2πt β . In particular, the fastest growing term, h 2 (2h+1) 2 32 e 4πt β in (3.27), naively seems to violate the bound on chaos because it grows exponentially with growth rate 4π β . However, it turns out that it reduces the Lyapunov exponent because its contribution to the OTOC has opposite sign to the leading exponential growth.
Note that each contribution of order O(g 4 ) (e.g., the analytic continuation of (3.18), (3.20), (3.21)) includes exponentially growing terms such as t 2 e 4πt β and te (3.29) One can easily see that the fastest exponential growth at of order O(g 4 ) increases the Lyapunov exponent.
To see the change of the Lyapunov exponent concretely, we go back to the original statement of the bound on chaos where we define the Lyapunov exponent by where F const corresponds to the constant terms 4 in F(t). The bound on chaos states  Figure 9: Lyapunov exponent λ L (t) from the total contribution to OTOC and from the loop contribution to OTOC. We plot them for the case of g = 1/100, 1/1000, 1/10000, and the corresponding scrambling time would be 2π β t ∼ log g 2 9.21, 13.82, 18.42.
where t * ∼ β 2π log 1 g 2 is the scrambling time. We plot numerically the Lyapunov exponent λ L (t) as a function of time from F OTOC (t) in (3.27) and from F OTOC, loop (t) in (3.29), respectively. See Figure 9. Here, we plot the Lyapunov exponent λ L (t) for g = 1/100, 1/1000, 1/10000 of which the scrambling time would be 2π β t ∼ log g 2 9.21, 13.82, 18.42, respectively. In Figure 9a, the Lyapunov exponent λ L (t) from F OTOC (t) in (3.27) is less than 2π β . As time increase, the Lyapunov exponent quickly saturates the bound, and it begins to decrease around the scrambling time. The term g 4 h 2 (2h+1) 2 32 e 4πt β at order O(g 4 ) in (3.27) is responsible for this decrease of Lyapunov exponent. The Lyapunov exponent seemingly vanishes beyond the scrambling time t * . However, we cannot trust the Lyapunov exponent beyond the scrambling time because the small g perturbation will break down.
On the other hand, if we had only the loop correction for the quantum correction to the OTOC, we would observe the violation of chaos bound before the scrambling time t * . See Figure 9b. This violation of the bound mainly comes from the fastest exponential growth −g 4 2h 2 β 2 t 2 e 4πt β at order O(g 4 ) in (3.29).

Conclusion
In this paper, we have evaluated the quantum correction of order O(g 4 ) by the Schwarzian soft mode to the OTOC. As is well known, the OTOC at order O(g 2 ) grows exponentially with the maximum growth rate 2π β [6,12,25]. At order O(g 4 ), we have found that the loop correction by the Schwarzian soft modes and the correction by two soft mode exchanges make the OTOC grow faster than the maximal growth. On the other hand, the correction by three soft mode scattering decreases the exponential growth rate. And, we have showed that the total correction slows down the exponential growth of the OTOC.
It is important to issue caveats in our analysis. First of all, we have not shown that the chaos bound would hold beyond the scrambling time, but we have found that the soft mode contribution to the OTOC at order O(g 4 ) slows down the exponential growth of order O(g 2 ).
To see the behavior of the OTOC beyond the scrambling time, one need to go beyond the perturbation, or, at least, the higher order corrections are required to estimate the behavior beyond the scrambling time. For example, O(g 6 ) might have e 6πt β growth, and depending on its sign the behavior around the scrambling time might be changed. We hope to report the higher order calculations in near future. Also, it would be highly interesting to find a constraint on the behavior of OTOC beyond the scrambling time from a simple physical argument.
In addition, our analysis is based on the Schwarzian theory which might not be as universal as other approaches such as the Virasoro conformal block [29][30][31] or pole-skipping phenomenon [53][54][55][56][57][58][59]. Nevertheless, the low energy physics of many interesting models such as SYK-like models and the dilaton gravity on nearly AdS 2 is described by Schwarzian action, in which our result can provide the understanding of the quantum correction to the chaos. It is interesting to explore the quantum correction to chaos in the context of "pole-skipping" phenomenon in the CFT 2 or higher dimensional CFT.
Finally, we have not evaluated all of the OTOC at order O(g 4 ), but we have calculated the contribution of the Schwarzian soft mode at order O(g 4 ). We have not considered the interaction of the matter fields which could also give a contribution to the OTOC. Unlike the soft mode, the contribution of matter fields might not be universal, but it would possibly depend on the details of models. Nevertheless, it would be interesting if the quantum correction to the chaos can constraint the matter interaction.