Quantum Geometry in Hyperbolic Band Theory

Hyperbolic band theory (HBT) extends condensed matter physics to negatively curved spaces, but measuring the quantum metric—a fundamental tensor governing topological phases and transport—has remained elusive. Conventional Euclidean methods fail because standard position operators do not exist in non-commutative hyperbolic geometries. This paper seeks to develop an experimentally viable technique to dynamically extract the quantum metric within the Abelian sector of HBT.

The authors leverage the fact that while the physical hyperbolic lattice is curved, its Abelian Bloch states are parameterized by a flat Jacobian torus. By identifying these parameters with Aharonov-Bohm holonomies around non-contractible cycles of the hyperbolic surface, they bypass the need for local position operators. They introduce 'holonomy shaking,' a protocol that modulates these boundary holonomies via time-periodic hopping phases. This approach allows them to map the quantum metric to measurable excitation rates in finite periodic clusters.

The study validates the holonomy shaking protocol using numerical simulations on {8,3} regular and Kagome-like hyperbolic lattices. The authors demonstrate that the dynamically extracted metric components align with analytical predictions across the Jacobian torus. Furthermore, they show that the protocol successfully resolves both rank-one Abelian metrics and higher-rank non-Abelian metrics (in the case of degenerate flat bands). The residual errors in the extraction are shown to be physically motivated, clustering near symmetry-enforced band degeneracies where the metric itself diverges.

This work provides a robust blueprint for probing quantum geometry in hyperbolic matter. By establishing a bridge between abstract hyperbolic band theory and measurable dynamical observables, the authors pave the way for experimental investigations into exotic phenomena, such as superfluidity in interacting flat-band hyperbolic systems and spin-gravity coupling, which are otherwise difficult to characterize in non-Euclidean synthetic matter platforms.