A new universal ratio in Random Matrix Theory and quantum analog of Kolmogorov-Arnold-Moser theorem in Type-I and Type-II hybrid Sachdev-Ye-Kitaev models.

We investigate possible quantum analog of Kolmogorov-Arnold-Moser (KAM) theorem 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 introduce a new universal ratio which is the ratio of the next nearest neighbour (NNN) energy level spacing to characterize the KAM theorem. 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 level. 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. After defining the quantum analog of the KAM theorem in the context of RMT, we show that the KAM theorem holds when $ J/K \sim \sqrt{N} e^{-N} $ and shrinks exponentially fast in the thermodynamic limit. In the GOE or GSE case in type-I, we find a lower bound of the dual form of the KAM theorem which states the stability of quantum chaos. While the stability of quantum chaos in all the other cases are much more robust than the KAM theorem in the integrable side. 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 difference between the quantum phase transition (QPT) characterized by renormalization groups at $ N=\infty $, $ 1/N $ expansions at a finite $ N $, and the CIT characterized by the RMT at a finite $ N $.