Complexity growth of operators in the SYK model and in JT gravity

TL;DR

The paper studies chaotic operator growth by comparing a microscopic K-complexity in the SYK model to holographic complexity via the CV conjecture in JT gravity for PETS. It demonstrates that both approaches exhibit an exponential growth regime transitioning to linear growth after the scrambling time, with the crossover governed by Lyapunov dynamics and system size. It establishes quantitative connections among operator size, K-complexity, and holographic complexity, including near-equality of early-time growth in the conformal limit and linear late-time growth with slopes tied to energy scales. Finite-temperature effects, insertion angles, and operator heaviness are analyzed, revealing how gravitational backreaction and SL(2) charges shape the complexity-size correspondence across boundary and bulk pictures.

Abstract

The concepts of operator size and computational complexity play important roles in the study of quantum chaos and holographic duality because they help characterize the structure of time-evolving Heisenberg operators. It is particularly important to understand how these microscopically defined measures of complexity are related to notions of complexity defined in terms of a dual holographic geometry, such as complexity-volume (CV) duality. Here we study partially entangled thermal states in the Sachdev-Ye-Kitaev (SYK) model and their dual description in terms of operators inserted in the interior of a black hole in Jackiw-Teitelboim (JT) gravity. We compare a microscopic definition of complexity in the SYK model known as K-complexity to calculations using CV duality in JT gravity and find that both quantities show an exponential-to-linear growth behavior. We also calculate the growth of operator size under time evolution and find connections between size and complexity. While the notion of operator size saturates at the scrambling time, our study suggests that complexity, which is well defined in both quantum systems and gravity theories, can serve as a useful measure of operator evolution at both early and late times.

Figures (6)

• Figure 1: A cartoon of a Heisenberg operator inserted at the half of the thermal circle. Some gates of $U(t)^\dag$ and $U(t)$ on the unaffected channels have been canceled with each other, due to the switchback effect.
• Figure 2: (a) The Lanczos coefficient of the SYK model. The curves from the bottom to the top correspond to $q=4$ and $N=20,22,24,26,28,30$, respectively. (b) The plateau value of the Lanczos coefficient at different $N$. $\bar{b}$ given by the plateau value of the Lanczos coefficient $n > N/q$. The dashed line is fitted by a linear function, showing that the platue value is proportional to $N$. (c) The K-complexity of the Heisenberg operator $\sqrt2 \psi_1 (t)$ in the SYK model. We use the parameters $q=4$ and $N=30$. (d) The time derivative of the K-complexity shown in (c). The scrambling time is denoted by $t_\ast$. Due to the small system size, the exponential growth is not obvious.
• Figure 3: (a) The Lanczos coefficient of the SYK model at different temperatures. The slope of the solid line is determined by $2\alpha(\beta)$. We use the parameters $N=30, q=4, \mathcal{J}=1/\sqrt2$. (b) The K-complexity growth of a single Majorana fermion at different temperature.
• Figure 4: (a) A schematic trajectory of boundary particles in hyperbolic space. The two segments and the vertical line represent the world lines of the two boundary particles and the inserted particle. The true horizon is point $H_R$. (b) The configuration of the dilaton field in the global coordinate of $AdS_2$ with the operator insertions. The parameters are $Q=100$, $L=30$, $\mu=20$ and $\theta=\pi/20$. We plot the geodesics of $C_V(-t_d,t_d)$ (dashed) and $C_V(t_d,t_d)$ (dotted). These geodesics intersect the left boundary, left-outer horizon, left-inner horizon, right-inner horizon, right-outer horizon, and right boundary at points $\left\{Y_L,\,H_L^+,\,H_L^-,\,H_R^-,\,H_R^+,\,Y_R\right\}$.
• Figure 5: (a) The complexity growth of a light Heisenberg operator, where the contributions from the intervals between points $\left\{Y_L,\,H_L^+,\,H_L^-,\,H_R^-,\,H_R^+,\,Y_R\right\}$ are shown, where the contribution from $D(H_L^-,H_R^-)$ is too small to be seen. (b) The time derivative of the complexity growth. The two dashed lines indicate the dissipation time $t_d$ and the scrambling time $t_\ast$ in (\ref{['timescale']}). The parameters are $Q=100$, $L=30$, $\theta=0$, and $\mu=0.1$.