Walsh量子化面包师映射的量子方差与涨落
Quantum variance and fluctuations for Walsh-quantized baker's maps
AI总结：
该研究证明Walsh量子化面包师映射（除D=4外）的矩阵元涨落渐近高斯，方差由经典关联决定，并给出ETH版本，揭示本征态微观关联区别于Haar随机向量。
摘要：
Walsh量子化的面包师映射是环面上量子混沌的模型。我们证明，对于所有面包师映射缩放因子$D\ge2$（除$D=4$外），在典型情况下（在简并的本征空间上的Haar测度意义下），随机本征基$\{φ^{(j)}\}_{j=1}^{N}$的缩放矩阵元涨落$\sqrt{N}\{\langle φ^{(j)}|\operatorname{Op}_{k,\ell}(a)|φ^{(j)}\rangle-\int_{\mathbb{T}^2}a\}_{j=1}^{N}$的经验分布在半经典极限$N\to\infty$下渐近服从高斯分布，其方差由经典面包师映射的关联给出。这确定了这些本征基的量子遍历定理中的精确收敛速率。我们获得了这些本征态的本征态热化假设（ETH）的一个版本，包括非对角矩阵元的极限复高斯分布，其方差也由经典关联给出。经典关联的存在突显了这些本征态虽然随机，但具有微观关联，使其区别于Haar随机向量。对于唯一的$D=4$情况，矩阵元涨落的高斯性取决于经典可观测在环面分形子集上的取值。
英文摘要：
The Walsh-quantized baker's maps are models for quantum chaos on the torus. We show that for all baker's map scaling factors $D\ge2$ except for $D=4$, typically (in the sense of Haar measure on the eigenspaces, which are degenerate) the empirical distribution of the scaled matrix element fluctuations $\sqrt{N}\{\langle φ^{(j)}|\operatorname{Op}_{k,\ell}(a)|φ^{(j)}\rangle-\int_{\mathbb{T}^2}a\}_{j=1}^{N}$ for a random eigenbasis $\{φ^{(j)}\}_{j=1}^{N}$ is asymptotically Gaussian in the semiclassical limit $N\to\infty$, with variance given in terms of classical baker's map correlations. This determines the precise rate of convergence in the quantum ergodic theorem for these eigenbases. We obtain a version of the Eigenstate Thermalization Hypothesis (ETH) for these eigenstates, including a limiting complex Gaussian distribution for the off-diagonal matrix elements, with variances also given in terms of classical correlations. The presence of the classical correlations highlights that these eigenstates, while random, have microscopic correlations that differentiate them from Haar random vectors. For the single value $D=4$, the Gaussianity of the matrix element fluctuations depends on the values of the classical observable on a fractal subset of the torus.
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