On Chord Dynamics and Complexity Growth in Double-Scaled SYK

TL;DR

This paper analyzes the dynamics of the double-scaled SYK model using chord Hamiltonians, deriving analytic expressions for operator evolution, correlation functions, and the effects of temperature, with implications for bulk-boundary correspondence and gravity duals.

Contribution

It provides new analytic results for operator dynamics and correlation functions in the double-scaled SYK model, including semi-classical behaviors and finite-temperature effects.

Findings

Derived analytic profiles for bulk states in the chord Hilbert space.

Established the relation between chord number generating function and four-point correlators.

Showed operator spreading slows at lower temperatures.

Abstract

We study the time evolution governed by the two-sided chord Hamiltonian in the double-scaled SYK model, which induces a probability distribution over operators in the double-scaled algebra. Through the bulk-to-boundary map, this distribution translates into dynamical profiles of bulk states within the chord Hilbert space. We derive analytic expressions for such profiles, valid across a broad parameter range and all time scales. Additionally, we demonstrate how distinct semi-classical behaviors emerge by localizing within specific energy regions in the semi-classical limit. We revisit the doubled Hilbert space formalism as an isometric map between the one-particle sector of the chord Hilbert space and the doubled zero-particle sector. Utilizing this map, we obtain analytic results for correlation functions and investigate the dynamical evolution for chord operators. Specifically, we establish an equivalence between the chord number generating function in presence of matter chords and the crossed four-point correlation function, the latter is closely related to the $6j$-symbol of $U_{\sqrt{q}}(\mathfrak{su}(1,1))$. We also explore finite-temperature effects, showing that operator spreading slows as temperature decreases. In the semi-classical limit, we perform a saddle point analysis and incorporate the one-loop determinant to derive the normalized time-ordered four-point correlation function at infinite temperature. The leading correction reproduces the \(1/N\) connected contribution observed in the large-\(p\) SYK model. Finally, we examine the time evolution of total chord number in presence of matter in the triple-scaled regime, linking it to the renormalized two-sided length in JT gravity with matter.