Probing the Chaos to Integrability Transition in Double-Scaled SYK

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.