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

Understand

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.