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.

具有定制的站点间隧道和现场能量的工程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支持任何非零本征能的无限退化本征态。这种类似 Efimov 的状态表现出离散的标度对称性，并暗示着一种前所未有的使用双曲曲面研究量子反常的装置。此外，所有本征态都以指数方式定位在双曲坐标中，这标志着厄米特系统中量子漏斗效应的第一个例子。因此，任何初始波包都朝着庞加莱半平面或庞加莱圆盘上的等效物的边缘传播，提供一种有效的方案来收集二维的光和原子。我们的发现揭示了双曲空间的有趣特性，并表明 Efimov 状态可以被视为具有额外维度的弯曲空间的投影。