非厄米准晶中的对数关联势场与局域化
Logarithmically Correlated Landscapes and Localization in Non-Hermitian Quasicrystals
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中文总结 AI 辅助
该研究分析了经零准周期调制的非互易Hatano-Nelson环，推导其势场特性，揭示了刚度阈值决定局域化与临界多重分形相的转变，且机制可应用于非互易拓扑电路。
中文摘要 AI 辅助
我们研究了一维Hatano-Nelson环，其非互易跳变通过零进行准周期调制。规范变换将每个本征态映射到单个空间包络，一旦跳变沿准周期轨道消失，其对数就成为确定性的对数关联场。我们推导了精确的傅里叶表示，并证明势场方差随系统大小对数增长，其刚度由调制功率和不可公度频率的算术性质决定。扩展相在由Jensen公式给出的代数边界处终止。在奇异区域内，局域化要求刚度超过临界阈值：当刚度高于该阈值时，波函数权重集中在少数最高势场峰上，状态为局域化；当刚度低于该阈值时，权重扩散到过多竞争峰上，状态为临界多重分形。极值统计呈反常特性：峰间距随秩的对数的幂次增长，其指数编码了频率的连分数类型，区别于安德森、 Aubry-André和随机规范普适类。刚度在多带晶格中跨通道叠加，且所有特征在百分量级的组分无序下仍能保留，这使得该机制可应用于非互易拓扑电路。
英文摘要
We study a one-dimensional Hatano-Nelson ring whose nonreciprocal hopping is quasiperiodically modulated through zero. A gauge transformation maps every eigenstate onto a single spatial envelope, whose logarithm becomes a deterministic, logarithmically correlated field once the hopping vanishes along the quasiperiodic orbit. We derive an exact Fourier representation and prove that the landscape variance grows logarithmically with system size, with a stiffness set by the modulation power and the arithmetic of the incommensurate frequency. The extended phase terminates at an algebraic boundary given by Jensen's formula. Inside the singular regime, localization requires the stiffness to exceed a critical threshold: above it, the wavefunction weight concentrates on the few highest landscape peaks and the state is localized; below it, the weight spreads over too many competing peaks and the state is a critical multifractal. The extreme-value statistics are anomalous: peak gaps grow as a power of the logarithm of rank, with an exponent that encodes the continued-fraction type of the frequency, distinct from the Anderson, Aubry-André, and random-gauge universality classes. The stiffness adds across channels in multiband lattices, and all signatures survive percent-level component disorder, placing the mechanism within reach of nonreciprocal topolectrical circuits.
