Quantum Geometry in Hyperbolic Band Theory
Abstract
Hyperbolic band theory (HBT) has recently unveiled a wealth of exotic physical phenomena in negatively curved spaces. Concurrently, the quantum metric has emerged as a fundamental tensor governing quantum geometry and topological phases. In this Letter, we bridge these two frontiers by investigating the quantum metric in HBT. Because conventional Euclidean metric extraction protocols fundamentally fail in these non-commutative geometries, we introduce holonomy shaking, a novel, experimentally viable technique utilizing periodic lattice driving to dynamically extract the metric. Using the {8,3} regular hyperbolic and kagome-like lattices as concrete examples, we demonstrate the high-fidelity dynamical extraction of the quantum metric. Our results establish a robust blueprint for probing hyperbolic quantum geometries, paving the way for the discovery of exotic topological phenomena in hyperbolic quantum matter.
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BibTeX
@misc{mahmoud2026quantum,
title = {{Quantum Geometry in Hyperbolic Band Theory}},
author = {Ahmed Adel Mahmoud and Riccardo Sorbello and Ronny Thomale and Steven Rayan},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14876},
eprint = {2609.14876},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14876},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Quantum Geometry in Hyperbolic Band Theory
AU - Ahmed Adel Mahmoud
AU - Riccardo Sorbello
AU - Ronny Thomale
AU - Steven Rayan
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.14876
UR - https://arxiv.org/abs/2609.14876
AB - Hyperbolic band theory (HBT) has recently unveiled a wealth of exotic physical phenomena in negatively curved spaces. Concurrently, the quantum metric has emerged as a fundamental tensor governing quantum geometry and topological phases. In this Letter, we bridge these two frontiers by investigating the quantum metric in HBT. Because conventional Euclidean metric extraction protocols fundamentally fail in these non-commutative geometries, we introduce holonomy shaking, a novel, experimentally viable technique utilizing periodic lattice driving to dynamically extract the metric. Using the {8,3} regular hyperbolic and kagome-like lattices as concrete examples, we demonstrate the high-fidelity dynamical extraction of the quantum metric. Our results establish a robust blueprint for probing hyperbolic quantum geometries, paving the way for the discovery of exotic topological phenomena in hyperbolic quantum matter.
ER -