Curved Geometry in Quantum Matter: Hyperbolic Lattices and Emergent Wormholes

Curved geometry can profoundly reshape quantum dynamics, yet it can enter quantum systems in very different ways. This talk explores two complementary manifestations of geometry in quantum matter: the influence of intrinsic negative curvature on electronic states, and the emergence of effective curved-space dynamics from a spatially varying magnetic background.

The first part focuses on Dirac fermions on hyperbolic lattices, where finite negative curvature makes parallel transport nontrivial and introduces the spin connection as an essential ingredient of the low-energy theory. Its lattice implementation through geodesic Wilson-line factors leads to characteristic modifications of the low-energy spectrum and density of states.

The second part considers a complementary setting in which the geometry is not imposed externally. For an electron strongly coupled to a magnetic vortex, spatial variations of the local magnetization modify the orbital dynamics in a way that can be described by an effective curved metric. Remarkably, the resulting geometry takes the form of an Ellis-type wormhole, giving rise to characteristic geodesic motion and lensing of electronic wave packets.

Together, these examples illustrate a broader interplay between geometry and quantum matter: curved backgrounds can reshape quantum states and dynamics, while spatial structure within a material can give rise to effective geometries that govern quasiparticle motion.