A Random Matrix Model Towards the Quantum Chaos Transition Conjecture

Consider D random systems that are modeled by independent \(N\times N\) complex Hermitian Wigner matrices. Suppose they are lying on a circle and the neighboring systems interact with each other through a deterministic matrix A. We prove that in the asymptotic limit \(N\rightarrow \infty \), the whole system exhibits a quantum chaos transition when the interaction strength \(\Vert A\Vert _{{\textrm{HS}}}\) varies. Specifically, when \(\Vert A\Vert _{{\textrm{HS}}}\ge N^{{\varepsilon }}\), we prove that the bulk eigenvalue statistics match those of a \(DN\times DN\) GUE asymptotically and each bulk eigenvector is approximately equally distributed among the D subsystems with probability \(1-\textrm{o}(1)\). These phenomena indicate quantum chaos of the whole system. In contrast, when \(\Vert A\Vert _{{\textrm{HS}}}\le N^{-{\varepsilon }}\), we show that the system is integrable: the bulk eigenvalue statistics behave like D independent copies of GUE statistics asymptotically and each bulk eigenvector is localized on only one subsystem. In particular, if we take \(D\rightarrow \infty \) after the \(N\rightarrow \infty \) limit, the bulk statistics converge to a Poisson point process under the DN scaling.