Universal ratio in random matrix theory and chaotic-to-integrable transition in type-I and type-II hybrid Sachdev-Ye-Kitaev models

Abstract

We investigate chaotic to integrable transition in two types of hybrid SYK models, which contain both $q=4$ SYK with interaction $J$ and $q=2$ SYK with an interaction $K$ in type-I or ${(q=2)}^{2}$ SYK with an interaction $\sqrt{K}$ in type-II. These models include hybrid Majorana fermion, complex fermion, and bosonic SYK. For the Majorana fermion case, we discuss both $N$ even and $N$ odd case. We make exact symmetry analysis on the possible symmetry class of both types of hybrid SYK in the 10-fold way by random matrix theory (RMT) and also work out the degeneracy of each energy levels. We introduce a new universal ratio, which is the ratio of the next-nearest-neighbor (NNN) energy level spacing to characterize the RMT. We perform exact diagonalization to evaluate both the known NN ratio and the new NNN ratio, then use both ratios to study chaotic to integrable transitions (CIT) in both types of hybrid SYK models. Some preliminary results on possible quantum analog of Kolmogorov-Arnold-Moser (KAM) theorem and its dual version in the quantum chaotic side are given. We explore some intrinsic connections between the two complementary approaches to quantum chaos: the RMT and the Lyapunov exponent by the $1/N$ expansion in the large $N$ limit at a suitable temperature range. We stress the crucial differences between the quantum phase transition (QPT) characterized by renormalization groups at $N=\ensuremath{\infty}$, $1/N$ expansions at a finite $N$, and the CIT characterized by the RMT at a finite $N$: the former focus on the ground state and its low-energy excitations (edge states in the Fock space), the latter on excited states (bulk states in the Fock space). We also discuss eigenstate thermalization hypothesis (ETH)'s power on a quantum chaotic bulk state and its inability to encode the edge states. Comments on some previously related works are given. Some future perspectives, especially the failure of the Zamoloddchikov's c-theorem in 1d CFT RG flow are outlined.