Correlated disorder in the SYK <sub>2</sub> model
Pak Hang Chris Lau, Chen-Te Ma, Jeff Murugan, Masaki Tezuka
Abstract
Abstract We study the SYK 2 model of N Majorana fermions with random quadratic interactions through a detailed spectral analysis and by coupling the model to two- and four-point sources. In particular, we define the generalized spectral form factor (SFF) and level spacing distribution function by generalizing from the partition function to the generating function. For N = 2, we obtain an exact solution of the generalized SFF. It exhibits qualitatively similar behavior to the higher N case with a source term. The exact solution helps understand the behavior of the generalized SFF. We calculate the generalized level spacing distribution function and the mean value of the adjacent gap ratio defined by the generating function. For the SYK 2 model with a four-point source term, we find a Gaussian unitary ensemble behavior in the near-integrable region of the theory, which indicates a transition to chaos. This behavior is confirmed by the connected part of the generalized SFF with an unfolded spectrum. The departure from this Gaussian random matrix behavior as the relative strength of the source term is increased is consistent with the observation that the four-point source term alone, without the SYK 2 couplings turned on, exhibits an integrable spectrum from the SFF and level spacing distribution function in the large N limit.