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Engineering lattice models with tailored inter-site tunnelings and onsiteenergies could synthesize essentially arbitrary Riemannian surfaces with highlytunable local curvatures. Here, we point out that discrete synthetic Poincar\'ehalf-planes and Poincar\'e disks, which are created by lattices in flat planes,support infinitely degenerate eigenstates for any nonzero eigenenergies. SuchEfimov-like states exhibit a discrete scaling symmetry and imply anunprecedented apparatus for studying quantum anomaly using hyperbolic surfaces.Furthermore, all eigenstates are exponentially localized in the hyperboliccoordinates, signifying the first example of quantum funneling effects inHermitian systems. As such, any initial wave packet travels towards the edge ofthe Poincar\'e half-plane or its equivalent on the Poincar\'e disk, deliveringan efficient scheme to harvest light and atoms in two dimensions. Our findingsunfold the intriguing properties of hyperbolic spaces and suggest that Efimovstates may be regarded as a projection from a curved space with an extradimension.

Efimov-like states and quantum funneling effects on synthetic hyperbolic surfaces