分形基底上的设计量子态：紧致局域化、平带与边缘模式
Designer quantum states on a fractal substrate: compact localization, flat bands and the edge modes
AI总结：
本文在三角Sierpinski垫片分形基底上解析揭示紧致局域单粒子态，提出精确实空间重正化群消元方案，证明其数目在热力学极限下可至无穷，并填充周期阵列平带，结果印证二维光子波导网络实验观测。
摘要：
本文在以任意大尺寸的三角Sierpinski垫片建模的确定性分形基底上，，解析揭示并研究了紧致局域单粒子本我们在紧束缚形式体系内提出精确的实空间重正化群（RSRG）消元方案来分辨这些态，并论证若将分形基底扩展至其热力学极限，此类态的数目可为无穷。有趣的是，这些局域态最终会填充周期性排列的Sierpinski垫片图案中的非色散平带，无论图案多大皆然。我们的结果匹配并印证了近期在具有分形几何的二维光子波导网络上观测到的紧致局域平带态，并提供了其一个完整子集，原则上，这些子集在分形光子晶格实验中应可观测。
英文摘要：
Compact localized single particle eigenstates on a deterministic fractal substrate, modelled by a triangular Sierpinski gasket of arbitrarily large size, are unravelled and examined analytically. We prescribe an exact real space renormalization group (RSRG) decimation scheme within a tight binding formalism to discern these states, and argue that the number of such states can be infinite if the fractal substrate is enlarged to its thermodynamic limit. Interestingly, these localized states turn out to populate the non-dispersive, flat bands in a periodic array of Sierpinski gasket motifs, however large they may be. Our results match and corroborate the recently observed compact localized, flat band states engineered on two dimensional photonic waveguide networks with a fractal geometry, and provide a whole subset of them, which, in principle, should be observable in fractal photonic lattice experiments.
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