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

• URL: http://arxiv.org/abs/2604.14522v1
• Date: Thu, 16 Apr 2026 01:24:27 GMT
• Title: Double-scaled bosonic and fermionic embedded ensembles, complex SYK, and the dual Hilbert space
• Abstract summary: We derive the density of states along with the $2$- and $4$-point functions of embedded ensembles for both fermions and bosons in the double-scaled limit.<n>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.
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• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We derive the density of states along with the $2$- 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.

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