AI 中文总结
本研究针对恒定曲率表面受径向随机势与弱角向微扰的薛定谔粒子，通过角动量分解、单模投影等方法，揭示了曲率放大的角向局域化交叉行为，量化了单模投影的有效性。
AI 中文摘要
我们研究了恒定曲率表面上受径向随机势和较弱角度相关微扰作用的连续薛定谔粒子。精确的角动量分解将旋转对称问题简化为独立的一维随机径向通道。正的径向李雅普诺夫指数将径向安德森局域化与对称性施加的圆形概率分布区分开来。窄径向模式的投影产生了一个无序的一维环。对于物理弧长上平稳的角向无序，环的局域化长度随微扰振幅的反平方增长。将该长度与测地周长相等，得到壳层破裂半径，其在平坦空间中呈代数增长，而在曲率半径为$a$、角向无序振幅为$\boldsymbol{\u03b5}$的双曲表面上，增长形式为$2a\ln(1/\varepsilon)+O(1)$。转移矩阵和有限环计算确定了交叉系数。在与环模型相同的角能量下进行的直接极坐标网格计算，通过对六个独立径向无序实现的简并角动量子空间进行无偏追踪，量化了单模投影的有效性。不对易的弱无序极限和平曲率极限表明，这种刻意构造的各向异性无序系综存在曲率放大的交叉行为，而非双曲平面上的普适金属-绝缘体转变。
英文摘要
We study a continuum Schrödinger particle on a surface of constant curvature subject to a radial random potential and a weaker angle-dependent perturbation. Exact angular-momentum decomposition reduces the rotationally symmetric problem to independent one-dimensional random radial channels. A positive radial Lyapunov exponent distinguishes radial Anderson localization from the circular probability profile imposed by symmetry. Projection of a narrow radial mode produces a disordered one-dimensional ring. For angular disorder that is stationary in physical arc length, the ring localization length grows as the inverse square of the perturbation amplitude. Equating this length with the geodesic circumference yields a shell-breaking radius that grows algebraically in flat space but as $2a\ln(1/\varepsilon)+O(1)$ on a hyperbolic surface of curvature radius $a$ and angular-disorder amplitude $\varepsilon$. Transfer-matrix and finite-ring calculations determine the crossover coefficient. Direct polar-grid calculations at the same angular energy as the ring model use unbiased tracking of the degenerate angular-momentum subspace across six independent radial-disorder realizations and quantify the validity of the single-mode projection. The noncommuting weak-disorder and flat-curvature limits establish a curvature-amplified crossover for this deliberately anisotropic disorder ensemble, rather than a generic metal--insulator transition on the hyperbolic plane.
Comments13 pages, 9 figures
