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Consider $D$ independent random systems that are modeled by independent$N\times N$ complex Hermitian Wigner matrices. Suppose they are lying on acircle and the neighboring systems interact with each other through adeterministic matrix $A$. We prove that in the asymptotic limit $N\to \infty$,the whole system exhibits a quantum chaos transition when the interactionstrength $\|A\|_{HS}$ changes. More precisely, when $\|A\|_{HS}\geN^{\varepsilon}$, we prove that the bulk eigenvalue statistics match those of a$DN\times DN$ GUE asymptotically and each bulk eigenvector is equallydistributed among all $D$ subsystems with probability $1-o(1)$. These phenomenaindicate quantum chaos of the whole system. In contrast, when $\|A\|_{HS}\leN^{-\varepsilon}$, we show that the system is integrable: the bulk eigenvaluestatistics behave like $D$ independent copies of GUE statistics asymptoticallyand each bulk eigenvector is localized on only one subsystem. In particular, ifwe take $D\to \infty$ after the $N\to \infty$ limit, the bulk statistics of thewhole system converge to a Poisson point process under the $DN$ scaling.

A random matrix model towards the quantum chaos transition conjecture