Parisi's hypercube, Fock-space frustration and near-AdS$_2$/near-CFT$_1$ holography

Abstract

We consider a model of Parisi where a single particle hops on an infinite-dimensional hypercube, under the influence of a uniform but disordered magnetic flux. We reinterpret the hypercube as the Fock-space graph of a many-body Hamiltonian and the flux as a frustration of the return amplitudes in Fock space. We will identify the set of observables that have the same correlation functions as the double-scaled Sachdev-Ye-Kitaev (DS-SYK) model, and hence the hypercube model is an equally good quantum model for near-AdS$_2$/near-CFT$_{1}$ (NAdS$_2$/NCFT$_1$) holography. Unlike the SYK model, the hypercube Hamiltonian is not $p$ local. Instead, the SYK model can be understood as a Fock-space model with similar frustrations. Hence we propose this type of Fock-space frustration as the broader characterization for NAdS$_2$/NCFT$_1$ microscopics, which encompasses the hypercube and the DS-SYK models as two specific examples. We then speculate on the possible origin of such frustrations.

Figures (1)

• Figure 1: Left: a chord diagram contributing to $2^{-d}\left\langle {\text{Tr}} H^6\right\rangle$, which represents the hopping sequence $D_\nu D_\rho D _\mu D_\rho D_\nu D_\mu$. This diagram has a value of $q^2$. Right: a chord diagram contributing to a two-point insertion $2^{-d}\left\langle {\text{Tr}} H^2 O H^2 O\right\rangle$. The dashed line represents the $O$ chord and the solid lines represent the $H$ chords. This diagram has a value of $q \tilde{q}^2$.