On the relation between the magnitude and exponent of OTOCs

TL;DR

The paper analyzes early-time OTOC growth in large-N quantum systems using a dynamical mean-field framework and ladder diagrams. It derives a central ladder identity that connects the growth exponent $\varkappa$, the OTOC prefactor $C$, and a branching time $t_B$ through the retarded kernel, enabling computational shortcuts to obtain one quantity from another. Applications to large-$q$ SYK yield an explicit form for the prefactor $C$ in terms of a parameter $v$, while a 1D SYK-like chain reveals maximal chaos with $\varkappa = 1$ (i.e. $2\pi/\beta$) above a coupling threshold, accompanied by a wavefront velocity determined by the kernel’s pole structure. The authors also introduce stringy states to interpret commutator OTOCs as inelastic scattering, linking the OTOC formalism to an effective scattering picture and Schwarzian low-energy dynamics.

Abstract

We derive an identity relating the growth exponent of early-time OTOCs, the pre-exponential factor, and a third number called "branching time". The latter is defined within the dynamical mean-field framework, namely, in terms of the retarded kernel. This identity can be used to calculate stringy effects in the SYK and similar models, we also explicitly define "strings" in this context. As another application, we consider an SYK chain. If the coupling strength $βJ$ is above a certain threshold and nonlinear (in the magnitude of OTOCs) effects are ignored, the exponent in the butterfly wavefront is exactly $2π/β$.

Figures (4)

• Figure 1: Operators $X_1(\theta_1),\ldots,X_4(\theta_4)$ of complex times $\theta_j=\tau_j+it_j$ on the double Keldysh contour, with $\tau_1,\ldots,\tau_4$ alternating and evenly spaced: (a) The complex coordinate $z=e^{i\theta}$ is used so that points with positive real time $t$ are located inside the unit disk, $|z|=e^{-t}<1$; (b) The contour is drawn such that the real time goes to the left, which is convenient when acting by operators on left. (In this example, $t_1=t_2$ and $t_3=t_4=0$.)
• Figure 4: Graphic solution of the equations $\varkappa(p_1)=1$ and $v_s=i\varkappa(p_s)/p_s=i\varkappa'(p_s)$. The velocity $v_s$ is the slope of the red dashed line. In case (a) it is the actual butterfly velocity; in case (b) the butterfly velocity is equal to $v_1=1/|p_1|$, the slope of the blue line.
• Figure 5: Four regions with different OTOC behavior.
• Figure 7: Stringy state corresponding to a half of a retarded ladder.