Double-scaled bosonic and fermionic embedded ensembles, complex SYK, and the dual Hilbert space

TL;DR

This paper derives spectral properties and correlation functions of embedded ensembles for fermions and bosons in the double-scaled limit, establishing their equivalence to the double-scaled SYK model and introducing a dual Hilbert space framework.

Contribution

It extends the double-scaled universality class to include bosonic systems and develops a Wick product approach for computing spectral functions and correlations.

Findings

Embedded ensembles are equivalent to double-scaled SYK models.

Wick product simplifies the calculation of density of states and n-point functions.

Duality between moments and expectation values in the chord Hilbert space is established.

Abstract

We derive the density of states and $2$- and $4$-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 $n$-point functions directly in the energy basis. In this context, the Wick product is equivalent to normal ordering of $q$-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 $n$-point functions. Embedded ensembles are equivalent to complex SYK at fixed charge, and we show working directly with embedded ensembles streamlines the derivations.