Chaos on the Hypercube

We consider the hypercubic model originally introduced by Parisi as a model for an array of Josephson junctions. This is a model where the Hamiltonian is given by the discretized Laplacian on a d-dimensional hypercube with U(1) gauge fields on the links but with a magnetic flux of constant magnitude and random orientation through all faces. The spectral flow of this model resembles that of the Maldacena-Qi model, and at zero flux, it coincides with the Maldacena-Qi model at infinite coupling. It also has a ground state that is separated from the rest of the spectrum by a gap, and a discrete symmetry in additional to the bipartite chiral symmetry. As is the case for the SYK model, the spectral density of this model is given by density function of the Q-Hermite polynomials. We analyze the spectral correlations of this model and find that the spectral form factor and the number variance are in the universality class of the Gaussian Unitary Ensemble, while the eigenvalues near zero are described by the chiral Gaussian Unitary Ensemble.