From Chaos to Integrability in Double Scaled SYK via a Chord Path Integral

TL;DR

This work analyzes transitions between chaotic and integrable dynamics in a double-scaled SYK-like system described by chord diagrams. By formulating a coarse-grained path integral over chord densities and introducing saddle fields $g_n$ and $g_z$, the authors identify two competing saddle points corresponding to chaotic and quasi-integrable behavior, connected to the endpoints $\kappa=0$ and $\kappa=1$, with a first-order phase transition line that terminates at finite temperature. They develop multiple analytic approximations—expansions in $\kappa$, in $\nu$, and a high-temperature expansion—to obtain tractable expressions for the saddle-point fields and the action, and compute thermal correlators and Krylov exponents to illustrate drastic changes across the transition. The results provide a semi-classical generalization of the $G\Sigma$ framework to multi-Hamiltonian double-scaled systems, revealing a universal coupled Liouville description of chaos-to-integrability transitions with potential experimental relevance for engineered quantum systems.

Abstract

We study thermodynamic phase transitions between integrable and chaotic dynamics. We do so by analyzing models that interpolate between the chaotic double scaled Sachdev-Ye-Kitaev (SYK) and the integrable $p$-spin systems, in a limit where they are described by chord diagrams. We develop a path integral formalism by coarse graining over the diagrams, which we use to argue that the system has two distinct phases: one is continuously connected to the chaotic system, and the other to the integrable. They are separated by a line of first order transition that ends at some finite temperature.

Figures (4)

• Figure 1: A chord diagram with three $n$-chords and one $z$-chord. It contributes $\left(\frac{{\mathbb{J}}^2}{\lambda}\right)^4 \nu^6\kappa^2 q^3$ to the moment $m_8$.
• Figure 2: An illustration of the terms that are considered in each part of the coarse graining procedure.
• Figure 3: Comparison of the numerics vs leading order approximation for the action of the two phases.
• Figure 4: Phase diagram in the $\kappa-1/\beta \mathbb{J}$ plane. The red and yellow dots are first-order phase transition points. The former are in the low temperature region, and only they are used to obtain the power-law fit (blue). The first-order transition line terminates at the black dot.