Zhang Ren, Lv Chenwei, Yan Yangqian, Zhou Qi
School of Physics, Xi'an Jiaotong University, Xi'an 710049, China; Department of Physics and Astronomy, Purdue University, West Lafayette IN 47907, USA.
Department of Physics and Astronomy, Purdue University, West Lafayette IN 47907, USA.
Sci Bull (Beijing). 2021 Oct 15;66(19):1967-1972. doi: 10.1016/j.scib.2021.06.017. Epub 2021 Jun 19.
Engineering lattice models with tailored inter-site tunnelings and onsite energies could synthesize essentially arbitrary Riemannian surfaces with highly tunable local curvatures. Here, we point out that discrete synthetic Poincaré half-planes and Poincaré disks, which are created by lattices in flat planes, support infinitely degenerate eigenstates for any nonzero eigenenergies. Such Efimov-like states exhibit a discrete scaling symmetry and imply an unprecedented apparatus for studying quantum anomaly using hyperbolic surfaces. Furthermore, all eigenstates are exponentially localized in the hyperbolic coordinates, signifying the first example of quantum funneling effects in Hermitian systems. As such, any initial wave packet travels towards the edge of the Poincaré half-plane or its equivalent on the Poincaré disk, delivering an efficient scheme to harvest light and atoms in two dimensions. Our findings unfold the intriguing properties of hyperbolic spaces and suggest that Efimov states may be regarded as a projection from a curved space with an extra dimension.
设计具有定制的格点间隧穿和在位能量的晶格模型，可以合成具有高度可调局部曲率的基本上任意的黎曼曲面。在这里，我们指出，由平面中的晶格创建的离散合成庞加莱半平面和庞加莱圆盘，对于任何非零本征能量都支持无限简并的本征态。这种类埃菲莫夫态表现出离散的标度对称性，并暗示了一种使用双曲曲面研究量子反常的前所未有的装置。此外，所有本征态在双曲坐标中呈指数局域化，这意味着厄米系统中量子漏斗效应的首个例子。因此，任何初始波包都会朝着庞加莱半平面的边缘或其在庞加莱圆盘上的等效位置传播，提供了一种在二维中收集光和原子的有效方案。我们的发现揭示了双曲空间的有趣特性，并表明埃菲莫夫态可被视为来自具有额外维度的弯曲空间的投影。
