A Path Integral for Chord Diagrams and Chaotic-Integrable Transitions in Double Scaled SYK

TL;DR

The paper develops a robust path integral framework for chord diagrams with two species to study chaotic-to-integrable transitions in double-scaled SYK and p-spin systems. By coarse graining chord diagrams and deriving exact and semiclassical path integrals, it connects the dynamics to Liouville-type equations and yields a clear two-phase structure: a chaotic phase linked to the chaotic Hamiltonian and a quasi-integrable phase tied to the integrable one, separated by a low-temperature first-order transition. The methodology extends to generic Hamiltonians, enabling RG-flow-like analyses and rich phase diagrams that include zero-temperature transitions in certain parameter regimes. These results illuminate how chaos can be systematically controlled and mapped in holographic and many-body contexts, with potential experimental implications for programmable quantum simulators. The work provides a versatile toolkit—combining coarse-grained chord counting, transfer matrices, and Liouville dynamics—that can be applied to a broad class of double-scaled systems beyond DS-SYK and p-spin.

Abstract

We study transitions from chaotic to integrable Hamiltonians in the double scaled SYK and $p$-spin systems. The dynamics of our models is described by chord diagrams with two species. We begin by developing a path integral formalism of coarse graining chord diagrams with a single species of chords, which has the same equations of motion as the bi-local ($GΣ$) Liouville action, yet appears otherwise to be different and in particular well defined. We then develop a similar formalism for two types of chords, allowing us to study different types of deformations of double scaled SYK and in particular a deformation by an integrable Hamiltonian. The system has two distinct thermodynamic phases: one is continuously connected to the chaotic SYK Hamiltonian, the other is continuously connected to the integrable Hamiltonian, separated at low temperature by a first order phase transition. We also analyze the phase diagram for generic deformations, which in some cases includes a zero-temperature phase transition.

Figures (17)

• Figure 1: Phase diagrams in the $\kappa-1/\beta \mathbb{J}$ plane. The orange dots denote the numerically obtained first order transition points, and the orange line is the first order transition line. The black dot is where the first-order transition line terminates.
• Figure 2: A chord diagram that contributes to $m_6$, representing the Wick contractions of $\langle J_{I_1} J_{I_3}\rangle \langle J_{I_2} J_{I_5}\rangle \langle J_{I_4} J_{I_6}\rangle \mathop{\mathrm{Tr}}\nolimits(X_{I_1}\cdots X_{I_6})$. It contributes as $q^2$ to the sum as it has two chord intersections.
• Figure 3: A chord diagram that contributes to $\left\langle\mathop{\mathrm{Tr}}\nolimits\left(H^6\right)\right\rangle$, representing the Wick contractions of $\langle J_{I_1} J_{I_2}\rangle \langle B_{L_1} B_{L_3}\rangle \langle B_{L_2} B_{L_4}\rangle \mathop{\mathrm{Tr}}\nolimits(X_{I_1} Q_{L_1} X_{I_2} Q_{L_2} Q_{L_3} Q_{L_4})$. It has one $n$-chord and two $z$-chords, and contributes $\nu^2\kappa^4 q$ to the sum over diagrams.
• Figure 4: Illustration of the different steps in the coarse graining scheme.
• Figure 5: Illustration of the different steps in the coarse graining scheme for two types of chords.