Complexity Geometry and Schwarzian Dynamics

TL;DR

The paper shows that the low-energy dynamics of SYK-like systems, captured by the Schwarzian action, has a parallel in a simplified complexity geometry where geodesic motion on a hyperbolic disk reduces to a 1D particle in an exponential potential. This effective 1D description matches the Schwarzian/JT gravity dynamics for bulk distance, both in two-sided traversable wormholes and in one-sided black holes, and it naturally extends to the switchback effect and large-$q$ SYK regimes. By unifying these viewpoints, the work provides supporting evidence for the complexity = volume conjecture and reveals a universal, low-energy effective description of size/complexity in holographic systems. The results suggest that gravity-like emergent dynamics tied to quantum complexity persists beyond chaotic regimes and may be governed by simple 1D effective actions, with implications for holography and the interpretation of complexity in quantum many-body systems.

Abstract

A celebrated feature of SYK-like models is that at low energies, their dynamics reduces to that of a single variable. In many setups, this "Schwarzian" variable can be interpreted as the extremal volume of the dual black hole, and the resulting dynamics is simply that of a 1D Newtonian particle in an exponential potential. On the complexity side, geodesics on a simplified version of Nielsen's complexity geometry also behave like a 1D particle in a potential given by the angular momentum barrier. The agreement between the effective actions of volume and complexity succinctly summarizes various strands of evidence that complexity is closely related to the dynamics of black holes.

Figures (6)

• Figure 1: A particle on the hyperbolic disk. We indicate the centrifugal force (which is always repulsive).
• Figure 2: A particle on a positively curved space. The centrifugal force is sometimes attractive and sometimes repulsive.
• Figure 3: The 2-sided, uncoupled setup. At some later time, we can insert some matter, which kicks the Schwarzian boundaries.
• Figure 4: A 2-sided kick to the eternal traversable wormhole. Originally, the distance between the two sides is constant. The perturbation causes the boundaries to oscillate. This should be compared with Figure \ref{['sphere']}, where the complexity also oscillates after a perturbation.
• Figure 5: Summary of our quantitative match. We consider the length of the a 1-sided black hole (with an end of the world brane), potentially with a kick dictated by SL(2) charge conservation. This matches the complexity as calculated in the toy model. The change in trajectory is due to the additional angular momentum when a perturbation to the particle is applied. The length of the dashed red lines in both figures are to be identified.