Path integral for chord diagrams and chaotic-integrable transitions in double scaled SYK

We study transitions from chaotic to integrable Hamiltonians in the double scaled Sachdev-Ye-Kitaev (SYK) and <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>p</a:mi></a:math>-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 bilocal (<c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>G</c:mi><c:mi mathvariant="normal">Σ</c:mi></c:math>) 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. Published by the American Physical Society 2024