Keating J P, Marklof J, Williams I G
School of Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdom.
Phys Rev Lett. 2006 Jul 21;97(3):034101. doi: 10.1103/PhysRevLett.97.034101.
We develop a percolation model for nodal domains in the eigenvectors of quantum chaotic torus maps. Our model follows directly from the assumption that the quantum maps are described by random matrix theory. Its accuracy in predicting statistical properties of the nodal domains is demonstrated for perturbed cat maps and supports the use of percolation theory to describe the wave functions of general Hamiltonian systems. We also demonstrate that the nodal domains of the perturbed cat maps obey the Cardy crossing formula and find evidence that the boundaries of the nodal domains are described by stochastic Loewner evolution with diffusion constant close to the expected value of 6, suggesting that quantum chaotic wave functions may exhibit conformal invariance in the semiclassical limit.
我们为量子混沌环面映射本征向量中的节域开发了一种渗流模型。我们的模型直接源于量子映射由随机矩阵理论描述这一假设。对于受扰猫映射，证明了该模型在预测节域统计特性方面的准确性，并支持使用渗流理论来描述一般哈密顿系统的波函数。我们还证明了受扰猫映射的节域服从卡迪交叉公式，并发现证据表明节域边界由扩散常数接近预期值6的随机洛厄纳演化描述，这表明量子混沌波函数在半经典极限下可能表现出共形不变性。
