A new universal ratio in Random Matrix Theory and chaotic to integrable transition in Type-I and Type-II hybrid Sachdev-Ye-Kitaev models

TL;DR

This paper introduces a new universal ratio based on next-nearest neighbor energy level spacings to analyze chaotic to integrable transitions in hybrid SYK models, connecting Random Matrix Theory and quantum chaos.

Contribution

It proposes a novel NNN energy level ratio for characterizing quantum chaos transitions in hybrid SYK models, extending RMT analysis.

Findings

The new NNN ratio effectively distinguishes chaotic and integrable regimes.

Symmetry analysis classifies hybrid SYK models within the 10-fold way.

Preliminary links between RMT and quantum Lyapunov exponents are discussed.

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 neighbour (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. 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.