# Chaos on the hypercube

We analyze the spectral properties of a $d$-dimensional HyperCubic (HC) lattice model originally introduced by Parisi. The U(1) gauge links of this model give rise to a magnetic flux of constant magnitude $\phi$ but random orientation through the faces of the hypercube. The HC model, which also can be written as a model of $2d$ interacting Majorana fermions, has a spectral flow that is reminiscent of the Maldacena-Qi (MQ) model, and its spectrum at $\phi=0$, actually coincides with the coupling term of the MQ model. As was already shown by Parisi, at leading order in $1/d$ , the spectral density of this model is given by the density function of the Q-Hermite polynomials, which is also the spectral density of the double-scaled Sachdev-Ye-Kitaev model. Parisi demonstrated this by mapping the moments of the HC model to $Q$-weighted sums on chord diagrams. We point out that the subleading moments of the HC model can also be mapped to weighted sums on chord diagrams, in a manner that descends from the leading moments. The HC model has a magnetic inversion symmetry that depends on both the magnitude and the orientation of the magnetic flux through the faces of the hypercube. The spectrum for fixed quantum number of this symmetry exhibits a transition from regular spectra at $\phi=0$ to chaotic spectra with spectral statistics given by the Gaussian Unitary Ensembles (GUE) for larger values of $\phi$. For small magnetic flux, the ground state is gapped and is close to a Thermofield Double (TFD) state.

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