A random matrix model towards the quantum chaos transition conjecture

Abstract

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 $N\to \infty$, the whole system exhibits a quantum chaos transition when the interaction strength $\|A\|_{HS}$ varies. Specifically, when $\|A\|_{HS}\ge N^{\varepsilon}$, 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 $1-o(1)$. These phenomena indicate quantum chaos of the whole system. In contrast, when $\|A\|_{HS}\le N^{-\varepsilon}$, 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 $D\to \infty$ after the $N\to \infty$ limit, the bulk statistics converge to a Poisson point process under the $DN$ scaling.

Figures (3)

• Figure 1.1: The entries $|{\bf{v}}(k)|^2$ for $k\in \mathcal{I}$, where ${\bf{v}}$ is the $(DN/2)$-th eigenvector of $H_{\Lambda}$. We choose $D=10$, $N=200$, and $A = \lambda I$. In (a) and (b), the red lines mark the value $N^{-1}$; in (c) and (d), the red lines mark the value $(DN)^{-1}$.
• Figure 1.2: The bulk eigenvalue gap distributions under the same setting as \ref{['simfigvec']}. The normalized histograms plot the rescaled eigenvalue gaps $DN\rho_{sc}(\lambda_k)(\lambda_{k+1}-\lambda_k)$, where $\lambda_k$ are the eigenvalues of $H_{\Lambda}$ with $k\in \llbracket{DN/20,19DN/20}\rrbracket$ and $\rho_{sc}$ is the semicircle density given by \ref{['eq:semidensity']} below. In (a) and (b), the red curves plot the probability density function for the exponential distribution: $f(x)=e^{-x}$. In (c) and (d), the red curves plot the Wigner surmise: $f(x)=\frac{\pi x}{2}e^{-\pi x^2/4}$. Note that repulsions between eigenvalues already appear to be present in the $\lambda = N^{-0.6}$ case. We attribute this phenomenon to the finite-$N$ effect.
• Figure 1.3: A 2D quantum billiard inside a circle with a potential $V$ supported in a compact region. The left picture depicts an integrable system featuring a radially symmetric potential, which does not mix eigenstates with different angular momentum. In the right picture, the rotational symmetry of $V$ is broken, resulting in mixing of eigenstates with different angular momentum, which can induce chaotic behavior. Note when the potential $V\to \infty$, essentially forming an inner boundary, the system becomes a quantum billiard between two boundaries.

Theorems & Definitions (64)

• Remark 1.1
• Remark 1.2
• Remark 1.3
• Theorem 2.2: Chaotic regime: eigenvectors
• Theorem 2.3: Chaotic regime: eigenvalues
• Theorem 2.4: Integrable regime: eigenvectors
• Theorem 2.5: Integrable regime: eigenvalues
• Remark 2.6
• Remark 2.7
• Definition 2.8: Matrix limit of $G$