On Chord Dynamics and Complexity Growth in Double-Scaled SYK

TL;DR

This work develops analytic control over real-time chord dynamics in the double-scaled SYK model, revealing how a two-sided chord Hamiltonian induces a probability distribution over chord operators that maps to bulk wormhole states via a bulk-boundary dictionary. It builds a comprehensive framework for operator growth (Krylov complexity), including an isometric factorization between one-particle and zero-particle sectors, and connects chord-number statistics to crossed four-point functions and quantum-group 6j symbols. The analysis spans generic q, finite temperatures, and semi-classical limits, with detailed saddle-point and one-loop calculations that reproduce known large-N SYK behavior and link to JT gravity with matter. Finite-temperature effects slow operator spreading, while the triple-scaling limits reveal distinct bulk interpretations, from Gaussian center-dominated dynamics to Schwarzian low-energy physics at the spectrum edge. The paper also outlines promising extensions to supersymmetry, multi-particle states, and stringy bulk descriptions, offering a rich bridge between boundary Krylov dynamics and bulk gravitational observables.

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.

Figures (11)

• Figure 1: A plot of the solutions $\phi_0(t), \phi_1(t), \dots, \phi_4(t)$ from Eq. \ref{['eq:overlap-00']} is shown, where $q = 0.8$, and only the first 15 terms in the series are included. The plot displays the functional dependence on the rescaled time $\tilde{t}$. The discrepancy between this finite series approximation and the numerical results obtained through direct integration \ref{['eq:phi-def']} is illustrated in Fig. \ref{['fig:error']}.
• Figure 2: A plot of the discrepancy between the solutions $\phi_0(t), \dots, \phi_4(t)$ with $q=0.8$, using only the first 15 terms in the series from Eq. \ref{['eq:overlap-00']}, and the numerical integral of Eq. \ref{['eq:phi-def']}, is shown as a function of the rescaled time $\tilde{t}$. The values of $\tilde{t}$ are sampled at $0, 0.5, 1, \dots, 9.5, 10$, and the error is less than $10^{-13}$.
• Figure 3: The plot illustrates the Krylov complexity, $\mathcal{K}(t)$ as a function of time $t$ for various values of $q$. We truncate the sum in Eq. \ref{['eq:K-def']} to the first 50 terms, with $q$ ranging over $\{0, 0.2, 0.5, 0.8, 0.99\}$. The results show that the growth of complexity becomes faster as $q$ approaches 1.
• Figure 4: The figure shows the time dependence of $\langle\Delta;0,0|\Delta(t,-t)\rangle$, with $q=0.8$ and different values of $\Delta$ as a signature of operator spreading. The decay becomes faster with the increase of matter weight $\Delta$. In particular, in the no-particle case with $\Delta=0$ the result stays a constant for all time. The case with $\Delta=\infty$ shows fastest decaying.
• Figure 5: A plot of $\langle\Delta; 0,0|\Delta(t,t)\rangle$ with increasing matter weight $q^\Delta=1, 0.8,0.5, 0.2$ and $0$ respectively and $q=0.8$. The initial decay of the overlapping function slows down as one increases the matter weight.