Operator growth in the SYK model

TL;DR

The paper investigates how a simple fermion operator in the SYK model evolves into increasingly complex many-body operators by analyzing the full size distribution P_s(t). It recasts operator growth as a quantum walk on an expanding graph of operator basis elements and develops both numerical (N=30) and analytic (large-N, large-q) methods to compute P_s(t). The large-q analysis yields explicit leading-order expressions for P_s(t), relates the mean size to the infinite-temperature OTOC growth, and identifies the chaos exponent λ_L = 2𝒥 in that limit, with a resummed form incorporating t/q corrections. Together, these results illuminate scrambling dynamics and provide a concrete link between operator growth, OTOCs, and underlying graph dynamics, with potential holographic interpretations and avenues for finite-N and finite-temperature extensions.

Abstract

We discuss the probability distribution for the "size" of a time-evolving operator in the SYK model. Scrambling is related to the fact that as time passes, the distribution shifts towards larger operators. Initially, the rate is exponential and determined by the infinite-temperature chaos exponent. We evaluate the size distribution numerically for $N = 30$, and show how to compute it in the large-$N$ theory using the dressed fermion propagator. We then evaluate the distribution explicitly at leading nontrivial order in the large-$q$ expansion.

Figures (2)

• Figure 1: Exact diagonalization numerics for SYK with $N = 30$ and $q = 4$ (see § \ref{['sec:graph']} for the definition of this model). At left, we plot the distribution of sizes in the operator $\psi_1(t)$ as a function of time. Notice that in the early phase, the peaks occur more rapidly as time passes. This is because already-large operators can grow faster than small ones. The "scrambling time" where the operator reaches full size would fall somewhere around three-quarters of the way through the plot. At right, we plot both the mean value and the variance of the size.
• Figure 2: The graph of operators. In (a) we show the first four layers. Vertices correspond to basis operators, whose associated fan diagrams are indicated in blue. The problem of the time evolution of $\psi_1(t)$ in the large $N$ theory is equivalent to the motion of a quantum particle on this graph (extended to further layers). In (b), (c), and (d), we show versions of the graph where we limit the recursive depth of the fan diagrams. The return amplitude on these graphs gives the zeroth, first and second iterations of the real-time Schwinger-Dyson equations. For any finite cutoff these amplitudes oscillate in time, but for the infinite graph the return amplitude decays exponentially.