Probing the Chaos to Integrability Transition in Double-Scaled SYK

TL;DR

The paper investigates how a thermodynamic first-order phase transition affects dynamical chaos in the BBJM model, which interpolates between the double-scaled SYK and an integrable chord Hamiltonian. By analyzing chord number evolution, Krylov (spread) complexity, operator size, and OTOCs across the phase diagram, the authors show a jump in chord numbers at the transition but no sharp qualitative change in scrambling diagnostics, indicating that the thermodynamic kink does not force a decisive chaos-integrability transition. The work develops and applies a detailed chord-diagram and Liouville-theory framework to extract chaos signatures inside both phases and near the transition, including perturbative analyses around chaotic and integrable saddles. These results have implications for holographic interpretations, suggesting a nuanced relation between bulk geometry (e.g., geodesic lengths) and spread complexity, while highlighting that near-transition scrambling remains blurred between integrable and chaotic regimes. The findings clarify how thermodynamic transitions relate to dynamical chaos diagnostics in many-body quantum systems and outline future avenues (finite-N studies, local quenches, and multi-flavor extensions) to deepen holographic and experimental connections.

Abstract

We investigate how a thermodynamical first-order phase transition affects the dynamical chaotic behaviour of a given model. To this effect, we analyze the model of Berkooz, Brukner, Jia and Mamroud that interpolates between the double-scaled SYK model and an integrable chord Hamiltonian. This model displays a first-order phase transition given by a kink in the free energy. We map out the dynamical behaviour, as characterized by chord number, Krylov complexity, and operator size, of the model across the phase diagram. We observe a jump in the chord numbers at the transition point, in agreement with the first-order transition. We further determine how scrambling measures, i.e.~the growth of the Lanczos coefficients and the time dependence of the operator size, change across the phase diagram. Deep inside the two phases, these measures indeed display integrable and chaotic behaviour, respectively. Across the transition however, we observe no qualitative change in these measures. This means that the thermodynamical transition does not imply a sharp transition in the growth exponent characterizing the dynamical chaotic behaviour. We also discuss a possible holographic interpretation of the model.

Figures (8)

• Figure 1: Schematic representation of the phase transition found in Berkooz:2024ofm (Fig. 11 b) in the on-shell action \ref{['eq:S on']} at finite temperature. The blue, red and orange solid lines correspond to the chaotic, a subdominant, and quasi-integrable saddle respectively.
• Figure 2: Evolution of $l(t)$\ref{['eq:def l(t)']} along real time, at $\beta\mathcal{J}=0$ and $\kappa=0,0.2,0.4,0.6,0.8,1.0$. The integrable system limit ($\kappa=0$) displays the expected parabolic growth; while the chaotic one ($\kappa=1$) displays late time linear growth, as seen in \ref{['eq:int saddles']} and \ref{['eq:euclidean chaotic saddle']} with $\tau=\mathrm i t$ ($\beta=0$) respectively. Note that the blue, amber and green lines nearly overlap with each other.
• Figure 3: $-g_n(\beta/2)$ and $-g_z(\beta/2)$\ref{['eq:explicit gnz']} in the dominating saddle as functions of $\kappa$ at $\beta\mathcal{J}=100$. The dashed curve denotes the critical value $\kappa_c=0.18$. The crossing of $-g_n(\beta/2)$ and $-g_z(\beta/2)$ at $\kappa_c$ is a consequence of the first-order transition, which shows where the dominant saddle jumps from the chaotic to the quasi-integrable solution.
• Figure 4: Real time evolution of $l(t)=-\tilde{g}_n(t)-\tilde{g}_z(t)$ in different phases near the phase transition line at $\beta\mathcal{J}=100$ and $\kappa=0.18$. The solid blue curve is covered by the green curve and the orange dashed curve is covered by the green dashed curve.
• Figure 5: (Left) The Lanczos coefficients $b_n$ at $\beta\mathcal{J}=0$ for different values of $\kappa$ in the $\Delta\to0$ limit. (Right) Krylov complexity for matter chord operators for different values of $\kappa$ while keeping $\beta\mathcal{J}=0$. Note that the complexity of largest value of $\kappa$ already converges to the result in \ref{['eq:integrable Krylov']}. At $\beta \mathcal{J}=0$, the Lanczos coefficients and Krylov complexity interpolate from chaotic behaviour at small $\kappa$ to integrable behaviour at large $\kappa$. Note there is no phase order transition due to the infinite temperature limit. In the left figure, the blue, amber and green set of dots are nearly coincident, while on the right the blue and brown solid curves are nearly coincident with each other.