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 synthe...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 Poincaréhalf-planes and Poincarédisks,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 Poincaréhalf-plane or its equivalent on the Poincarédisk,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.展开更多
基金supported by the National Natural Science Foundation of China(11804268)the National Key R&D Program of China(2018YFA0307601)。
文摘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 Poincaréhalf-planes and Poincarédisks,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 Poincaréhalf-plane or its equivalent on the Poincarédisk,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.
