Emergent states and algebras from double-scaling limit of pure states in SYK

Recent work has highlighted a subtle feature of large-N limits in holography: a sequence of pure microscopic states need not remain pure with respect to the emergent algebra of observables. In this talk, I will discuss this phenomenon for Kourkoulou--Maldacena (KM) states in the double-scaling limit of the SYK model, and show that their ensemble-averaged algebraic description depends crucially on which observables are retained in the limit. For fermionic operators of size p ~ N^{1/2}, generic operators converge to the usual chord operators of double-scaled SYK. The resulting von Neumann algebra is the standard Type II_1 factor, and the KM pure states at infinite temperature converge to the tracial state. In this sense, generic probes lose access to microscopic purity. I will then explain how this conclusion changes once one includes a special class of operators adapted to the KM state. These operators survive the double-scaling limit as dressed chord creation and annihilation operators, with the dressing encoding the distance to the KM state through the Euclidean bulk. With these operators included, the limiting algebra becomes a Type I_\infty factor, and the sequence of KM states converges to a pure state. This provides a solvable example in which adding sufficiently state-adapted operators to the emergent algebra restores access to the purity of the underlying state.

Along the way, I will describe the modified chord rules governing the dressed operators and explain their correlation functions, semiclassical limit, and connections to JT gravity coupled to matter and end-of-the-world branes. I will also discuss a solvable deformation to the Hamiltonian in which bound states emerge above a critical threshold. I will end by drawing broader lessons for emergent algebras in holography, including possible analogies with black hole interiors and closed universes.