New families of toy universes stretch the SYK quantum gravity laboratory

Starting from the double-scaled SYK model — a tractable quantum system dual to a two-dimensional dilaton gravity — the authors build a tower of deformations governed by the q-Askey scheme of orthogonal polynomials. In the semiclassical limit, the chord number tracking operator complexity maps onto the length of a wormhole bridging an end-of-the-world brane to an AdS boundary. Cranking up one deformation parameter discretizes the energy spectrum and triggers a geometric phase transition in sine-dilaton gravity. The operator algebras of the deformed theories interpolate between type II₁ and type I∞ factors, and entanglement entropy between the type II₁ algebras reproduces the Ryu-Takayanagi formula through an extremal surface.