On Chord Dynamics and Complexity Growth in Double-Scaled SYK

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.