Operator Size for Holographic Field Theories

TL;DR

The paper defines a state-dependent operator size S_{| Psi angle}( O) as the expectation value of a positive semi-definite operator S_{| Psi angle}, and demonstrates its realization in fermionic SYK-like systems and in holographic field theories. In SYK, it reproduces the Qi–Streicher relation at leading order and extends to time-shifted TFDs and pure-state black holes, while in holographic CFTs the leading size is tied to the CFT Hamiltonian or Kruskal momenta, with bulk interpretations via shockwaves and eikonal phases. The work further proposes a bulk, backreaction-aware proposal for operator size using averaged eikonal phases, predicting exponential growth, a subsequent linear regime, and saturation at the black-hole entropy, and highlighting universality with entanglement and complexity at linear order. Together, these results illuminate a boundary–bulk size correspondence across diverse geometries (AdS_3, AdS-Rindler, BTZ) and offer a framework for studying operator growth beyond thermal ensembles, with potential implications for traversable wormholes and holographic complexity.

Abstract

We formulate a state-dependent definition of operator size that captures the effective size of an operator acting on a reference state. We apply our definition to the SYK model and holographic 2-dimensional CFTs, generalizing the Qi-Streicher formula to a large class of geometries which includes pure AdS$_3$ and BTZ black holes. In pure AdS$_3$, the operator size is proportional to the global Hamiltonian at leading order in $1/N$, mirroring the results of Lin-Maldacena-Zhao in AdS$_2$. For BTZ geometries, it is given by the sum of the Kruskal momenta. Higher $1/N$ corrections become relevant when backreaction gets large, and we expect a transition in the growth pattern that depends on the transverse profile of the excitation. We propose a bulk dual that captures this profile dependence and exhibits saturation at a size of order the black hole entropy. This bulk dual is an averaged eikonal phase over a class of scattering events, and it can be interpreted as the "number of virtual gravitons" in the gravitational field created by an infaller.

Figures (5)

• Figure 1: The pure state \ref{['PureSYKState']} is dual to a single-sided black hole with a ETW brane (red) behind the horizon (green). The off-diagonal 2-point functions are given by powers of the length of the geodesic that connects a point on the boundary to the "center" of the ETW brane.
• Figure 2: The AdS$_3$ vacuum can be expressed in as an entangled state of two line CFTs. In the bulk, this corresponds to using accelerating coordinates which produce the horizons which are shaded blue (the red line denotes the bifurcation surface). The green/blue arrows show the Rindler/global Hamiltonian evolution in the bulk respectively.
• Figure 3: The plot shows the typical growth structure of OTOCs $\langle O^\dagger_R(-t,0) V_L(0,x) V_R(0,x) O_R(-t,0) \rangle$. The velocity of this lightcone is the butterfly velocity, which in 2 dimensions is equal to the speed of light.
• Figure 4: As we push $\mathcal{O}(-t)$ to earlier times, it creates an increasingly strong time delay. The pulse that reaches the boundary at $t=0$ must have been emitted earlier than naively expected, thus reducing the relative boost between the pulse and the infalling particle.
• Figure 5: Comparison of the size growth for a localized excitation and a spherical shell with equal energy. We have chosen the parameters to make it easier to visualize the exponential-linear growth transition.