  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 Poincar\'e
half-planes and Poincar\'e 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 Poincar\'e half-plane or its equivalent on the Poincar\'e 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.

Quantum Gases;Quantum Physics