Double-scaled bosonic and fermionic embedded ensembles, complex SYK, and the dual Hilbert space
Abstract
We derive the density of states and - and -point functions of embedded ensembles for both fermions and bosons in the double-scaled limit. It is shown the models are equivalent to the double-scaled Sachdev-Ye-Kitaev model, expanding the double-scaled universality class to include both fermionic and bosonic systems. The models can be solved by introducing the Wick product of non-commuting Gaussian random variables. We show that deriving the Wick product is sufficient for computing the density of states, and properties of the Wick product can be used to compute -point functions directly in the energy basis. In this context, the Wick product is equivalent to normal ordering of -oscillators, which leads to the duality between moments of double-scaled models and expectation values in the chord Hilbert space. By considering operator probes as a second set of oscillators, we extend the duality to compute -point functions. Embedded ensembles are equivalent to complex SYK at fixed charge, and we show working directly with embedded ensembles streamlines the derivations.
Cite
@article{arxiv.2604.14522,
title = {Double-scaled bosonic and fermionic embedded ensembles, complex SYK, and the dual Hilbert space},
author = {Jarod Tall and Steven Tomsovic},
journal= {arXiv preprint arXiv:2604.14522},
year = {2026}
}
Comments
v2: minor edits. 24 pages + appendix